# On connected components with many edges

Sammy Luo

###### Abstract.

We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then
$H$ contains a connected component $H^{\prime}$ satisfying
$|E(H^{\prime})||E(G)|\geq|E(H)|^{2}$. We use this to prove that every three-
coloring of the edges of a complete graph contains a monochromatic connected
subgraph with at least $1/6$ of the edges. We further show that such a
coloring has a monochromatic circuit with a fraction $1/6-o(1)$ of the edges.
This verifies a conjecture of Conlon and Tyomkyn. Moreover, for general $k$,
we show that every $k$-coloring of the edges of $K_{n}$ contains a
monochromatic connected subgraph with at least
$\frac{1}{k^{2}-k+\frac{5}{4}}\binom{n}{2}$ edges.

Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
Supported by NSF GRFP Grant DGE-1656518. Email<EMAIL_ADDRESS>
##  1\. Introduction

A classical observation of Erdős and Rado is that in any two-coloring of the
edges of the complete graph $K_{n}$, one of the color classes forms a
connected graph. In [3], Gyárfás proves the following generalization of this
observation: For any $k\geq 2$, in every $k$-coloring of the edges of $K_{n}$,
there is a monochromatic connected component with at least $n/(k-1)$ vertices.
This observation has since been extended in numerous ways, such as by
replacing $K_{n}$ with a graph of high minimum degree [5] or with a nearly-
complete bipartite graph [2], or adding a constraint on the diameter of the
large monochromatic component [7]. See [4] for a survey of earlier work on the
subject.

The arguments used in this subject tend to focus on sparse spanning structures
like double stars. As such, there is a surprising lack of progress on the
corresponding question about edges in a monochromatic connected component.
That is, what is the largest value of $M=M(n,k)$ such that every $k$-edge-
coloring of $K_{n}$ has a monochromatic connected component with at least $M$
edges?

This question was raised by Conlon and Tyomkyn in [1], in the context of
determining the multicolor Ramsey numbers for trails (see Section 3.2 for the
relevant definitions). After showing that $M(n,2)=\frac{2}{9}n^{2}+o(n^{2})$,
they sketch a simple argument that shows
$M(n,k)\geq\frac{1}{16k^{2}}n^{2}+O(n)$ for all $k$; with a slightly more
careful analysis, their argument in fact yields
$M(n,k)\geq\frac{1}{4k^{2}}n^{2}+O(n)$. In the other direction, they examine a
construction of Gyárfás in [4] to show that
$M(n,k)\leq\frac{1}{2k(k-1)}n^{2}+O(n)$ for infinitely many values of $k$
(specifically, when $k-1$ is a prime power), conjecturing that this upper
bound is tight in the case $k=3$.

In this note, we improve the general lower bound on $M(n,k)$, as well as a
corresponding lower bound for the trail Ramsey problem, bringing it to
asymptotically within a factor $1-O\left(\frac{1}{k^{2}}\right)$ of the upper
bound for infinitely many values of $k$.

######  Theorem 1.

For any $k\geq 2$, in every $k$-coloring of the edges of $K_{n}$, there is a
monochromatic connected component with at least
$M(n,k)\geq\frac{1}{k^{2}-k+\frac{5}{4}}\binom{n}{2}$ edges.

By building on the overarching ideas in [4] and introducing some key new
insights, we manage to strengthen this lower bound in the case $k=3$ to prove
the tight lower bound conjectured by Conlon and Tyomkyn.

######  Theorem 2.

In every $3$-coloring of the edges of $K_{n}$, there is a monochromatic
component containing at least a sixth of the edges. That is,
$M(n,3)\geq\lceil\frac{1}{6}\binom{n}{2}\rceil$. Moreover, for $n$
sufficiently large, this bound is sharp.

While equality holds in this bound for sufficiently large $n$, there are, as
we will see from the proof, small values of $n$ for which
$M(n,3)>\lceil\frac{1}{6}\binom{n}{2}\rceil$. In particular, equality holds
for all $n\geq 18$, but not for $n=17$.

The key result we use is a new inequality that may be of independent interest.
Given a subgraph $H$ of a complete multipartite graph $G$, it relates the edge
counts of $G$ and $H$ to the largest edge count of a connected component
$H^{\prime}$ of $H$.

######  Theorem 3.

Let $G$ be a complete $r$-partite graph for some $r\geq 2$, and let $H$ be a
subgraph of $G$. Then $H$ contains a connected component $H^{\prime}$
satisfying

$|E(H^{\prime})|\geq\frac{|E(H)|^{2}}{|E(G)|}.$

In effect, this result settles the density analogue of the coloring question
of determining $M(n,k)$. Instead of partitioning the edges of a graph $G$ into
color classes, we are fixing a subgraph $H$ of $G$ with a given fraction
$\delta=\frac{|E(H)|}{|E(G)|}$ of its edges, and asking about the component of
$H$ with the most edges. Theorem 3 can then be restated as: if
$|E(H)|=\delta|E(G)|$, then $H$ contains a connected component $H^{\prime}$
with $|E(H^{\prime})|\geq\delta^{2}|E(G)|$. Equality can be attained
asymptotically when $\delta=\frac{1}{k}$ for any positive integer $k\geq 2$:
Let $V(G)=V_{1}\cup\cdots\cup V_{r}$ be an $r$-partition of $V(G)$, split each
$V_{i}$ into $k$ (roughly) equally sized vertex sets
$\\{V_{i,j}\\}_{j=1}^{k}$, and for $1\leq j\leq k$, let $H_{j}$ be the
subgraph of $G$ induced on $\bigcup_{i=1}^{r}V_{i,j}$. Then indeed,
$H=\bigcup_{j=1}^{k}H_{j}$ is a subgraph of $G$ whose components have edge
counts

$|E(H_{j})|=\frac{1}{k^{2}}|E(G)|+O(n)=\frac{|E(H)|^{2}}{|E(G)|}+O(n).$

The simplest case of Theorem 3, when $G=K_{n}$, already immediately implies an
improvement over previously known lower bounds on $M(n,k)$.

######  Corollary 4.

In every $k$-coloring of the edges of $K_{n}$, there is a monochromatic
connected component with at least $\frac{1}{k^{2}}\binom{n}{2}$ edges.

###### Proof.

In a $k$-coloring of the edges of $K_{n}$, one of the color classes has at
least $\frac{1}{k}\binom{n}{2}$ edges. Taking $H$ to be this color class in
Theorem 3 with $G=K_{n}$ yields Corollary 4. ∎

The proof of Theorem 2 requires a more detailed argument, but similarly
follows from this one case of Theorem 3, while the proof of Theorem 1 requires
the use of Theorem 3 in full generality.

In the next section, we give an elementary proof of Theorem 3. We then study
the coloring version of the problem, as well as the corresponding trail Ramsey
problem, in Section 3.

##  2\. Proof of Theorem 3

In the case $G=K_{n}$, the proof of Theorem 3 is very simple, but nevertheless
includes a key insight: Instead of taking a component that maximizes the
number of edges right away, we consider a component $H^{\prime}$ with maximum
average degree
$\bar{d}(H^{\prime})=\frac{|E(H^{\prime})|}{\frac{1}{2}|V(H^{\prime})|}$. Two
observations are crucial here: First, by the so-called _generalized mediant
inequality_ , this highest average degree must be at least the average degree
of the whole graph $H$, since

$\frac{|E(H)|}{\frac{1}{2}|V(H)|}=\frac{\sum|E(H_{i})|}{\frac{1}{2}\sum|V(H_{i})|},$

where the sums are over connected components $H_{i}$ of $H$, and the right
hand side is a generalized mediant of the average degrees of the individual
components. Second, the number of vertices of $H^{\prime}$ is at least one
more than its maximum degree, so $|V(H^{\prime})|\geq\bar{d}(H^{\prime})+1$.
Letting $\delta=\frac{|E(H)|}{|E(G)|}$, we then obtain the bound

$\displaystyle|E(H^{\prime})|$
$\displaystyle\geq\frac{1}{2}(\bar{d}(H^{\prime}))(\bar{d}(H^{\prime})+1)=\binom{\bar{d}(H^{\prime})+1}{2}$
$\displaystyle\geq\binom{\bar{d}(H)+1}{2}=\binom{\frac{2|E(H)|}{n}+1}{2}$
$\displaystyle=\binom{\delta(n-1)+1}{2}\geq\delta^{2}\binom{n}{2}$
$\displaystyle=\delta^{2}|E(G)|=\frac{|E(H)|^{2}}{|E(G)|},$

as desired.

The general case will use both of these observations in a modified setting.
Instead of the average degree, we will work with a slightly different quantity
whose denominator is a weighted vertex count; we will then, perhaps
counterintuitively, lower bound this weighted vertex count by the modified
analogue of the average degree, in order to obtain the bound we seek. The core
of our proof is the following general inequality.

######  Lemma 5.

If $a_{1},\dots,a_{r}$ and $b_{1},\dots,b_{r}$ are nonnegative real numbers,
then

$\left(\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)-\sum_{i=1}^{r}a_{i}b_{i}\right)^{2}\geq\left(\left(\sum_{i=1}^{r}a_{i}\right)^{2}-\sum_{i=1}^{r}a_{i}^{2}\right)\left(\left(\sum_{i=1}^{r}b_{i}\right)^{2}-\sum_{i=1}^{r}b_{i}^{2}\right).$

###### Proof.

The Cauchy-Schwarz inequality yields

$\displaystyle\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)-\sum_{i=1}^{r}a_{i}b_{i}\geq\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)-\sqrt{\left(\sum_{i=1}^{r}a_{i}^{2}\right)\left(\sum_{i=1}^{r}b_{i}^{2}\right)}.$

This last quantity is clearly nonnegative, since
$\left(\sum_{i=1}^{r}a_{i}\right)^{2}\geq\sum_{i=1}^{r}a_{i}^{2}$ and
$\left(\sum_{i=1}^{r}b_{i}\right)^{2}\geq\sum_{i=1}^{r}b_{i}^{2}$. So,

$\displaystyle\left(\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)-\sum_{i=1}^{r}a_{i}b_{i}\right)^{2}\geq\left(\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)-\sqrt{\left(\sum_{i=1}^{r}a_{i}^{2}\right)\left(\sum_{i=1}^{r}b_{i}^{2}\right)}\right)^{2}$
$\displaystyle=\left(\sum_{i=1}^{r}a_{i}\right)^{2}\left(\sum_{i=1}^{r}b_{i}\right)^{2}+\left(\sum_{i=1}^{r}a_{i}^{2}\right)\left(\sum_{i=1}^{r}b_{i}^{2}\right)-2\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)\sqrt{\left(\sum_{i=1}^{r}a_{i}^{2}\right)\left(\sum_{i=1}^{r}b_{i}^{2}\right)}$
$\displaystyle\geq\left(\sum_{i=1}^{r}a_{i}\right)^{2}\left(\sum_{i=1}^{r}b_{i}\right)^{2}+\left(\sum_{i=1}^{r}a_{i}^{2}\right)\left(\sum_{i=1}^{r}b_{i}^{2}\right)-\left(\sum_{i=1}^{r}a_{i}\right)^{2}\left(\sum_{i=1}^{r}b_{i}^{2}\right)-\left(\sum_{i=1}^{r}b_{i}\right)^{2}\left(\sum_{i=1}^{r}a_{i}^{2}\right)$
$\displaystyle=\left(\left(\sum_{i=1}^{r}a_{i}\right)^{2}-\sum_{i=1}^{r}a_{i}^{2}\right)\left(\left(\sum_{i=1}^{r}b_{i}\right)^{2}-\sum_{i=1}^{r}b_{i}^{2}\right),$

where the last inequality is an application of the AM-GM inequality. ∎

Given a graph $G$ and vertex sets $S,T\subseteq V(G)$, let

$e_{G}(S,T)=\\#\\{(s,t)\in S\times T:\>s\sim_{G}t\\}.$

We write $e(S,T)$ for $e_{G}(S,T)$ when the graph in question is unambiguous.
Let $G[S]$ denote the induced subgraph of $G$ on the vertex set $S$. Lemma 5
immediately implies the following.

######  Corollary 6.

Let $G$ be a complete multipartite graph. For any $S,T\subseteq V(G)$, we have

$e(S,T)^{2}\geq 4|E(G[S])||E(G[T])|.$

###### Proof.

Suppose that $G$ is $r$-partite. Let $V(G)=V_{1}\cup\cdots\cup V_{r}$ be a
partition of the vertices of $G$ into $r$ independent sets. Let
$a_{i}=|V_{i}\cap S|$ and $b_{i}=|V_{i}\cap T|$, so

$e(S,T)=\sum_{\begin{subarray}{c}1\leq i,j\leq r\\\ i\neq
j\end{subarray}}a_{i}b_{j}=\left(\sum_{i=1}^{r}a_{i}\right)\left(\sum_{i=1}^{r}b_{i}\right)-\sum_{i=1}^{r}a_{i}b_{i},$

while

$|E(G[S])|=\frac{1}{2}\left(\left(\sum_{i=1}^{r}a_{i}\right)^{2}-\sum_{i=1}^{r}a_{i}^{2}\right),\qquad|E(G[T])|=\frac{1}{2}\left(\left(\sum_{i=1}^{r}b_{i}\right)^{2}-\sum_{i=1}^{r}b_{i}^{2}\right).$

Then Lemma 5 indeed yields $e(S,T)^{2}\geq 4|E(G[S])||E(G[T])|$, as desired. ∎

###### Proof of Theorem 3.

Let $V=V(G)=V_{1}\cup\cdots\cup V_{r}$ be a partition of the vertices of $G$
into $r$ independent sets, let $H_{1},\dots,H_{k}$ be the connected components
of $H$, and let $V_{i,\ell}=V_{i}\cap H_{\ell}$. We have

$|E(H)|=\sum_{j=1}^{k}|E(H_{\ell})|,\qquad|V_{i}|=\sum_{\ell=1}^{k}|V_{i,\ell}|\>\text{
for all }i.$

For any subset $S\subseteq V$, define $f(S)=\frac{1}{2}e_{G}(S,V)$. Note that

$f(S)=\frac{1}{2}\sum_{\begin{subarray}{c}1\leq i,j\leq r\\\ i\neq
j\end{subarray}}|V_{i}\cap S||V_{j}|,$

so $f(S)$ can be viewed as a weighted vertex count for $S$, with the property
that

$\sum_{\ell=1}^{k}f(V(H_{\ell}))=\frac{1}{2}\sum_{\ell=1}^{k}e_{G}(V(H_{\ell}),V)=\frac{1}{2}e_{G}(V,V)=|E(G)|.$

Then by the generalized mediant inequality, for some $H^{\prime}=H_{\ell}$ we
have

$\frac{|E(H^{\prime})|}{f(V(H^{\prime}))}\geq\frac{\sum_{\ell=1}^{k}|E(H_{\ell})|}{\sum_{\ell=1}^{k}f(V(H_{\ell}))}=\frac{|E(H)|}{|E(G)|}.$

Now, Corollary 6 applied with $S=V(H^{\prime})$, $T=V$ yields
$f(V(H^{\prime}))^{2}\geq|E(G[V(H^{\prime})])||E(G)|\geq|E(H^{\prime})||E(G)|$,
so that

$\frac{|E(H)|}{|E(G)|}\leq\frac{|E(H^{\prime})|}{f(V(H^{\prime}))}\leq\sqrt{\frac{|E(H^{\prime})|}{|E(G)|}},$

which rearranges to the desired inequality. ∎

##  3\. Coloring problems

We can now apply Theorem 3 to the corresponding coloring problems. We have
already seen that applying Theorem 3 with $G=K_{n}$ readily yields the simple
lower bound on $M(n,k)$ given in Corollary 4. We now discuss how to improve
this lower bound, before turning to a closely related problem on monochromatic
trails and circuits.

###  3.1. Lower bound on $M(n,k)$ for general $k$

Our strategy for improving the lower bound on $M(n,k)$ is as follows: First,
assuming that no monochromatic component has too many edges, we show that in
the color with the highest density (say, red), we can upper bound the number
of vertices covered by any set of components (or else we can finish with an
average degree argument on the rest of the red components). Through a
smoothing argument, this yields an upper bound on the sum of squares of the
number of vertices in each red component, and thus a lower bound on the number
of edges in the complete multipartite graph $G$ formed by deleting from
$K_{n}$ all edges within the vertex sets of the red components. We finish by
applying Theorem 3 to $G$ to find a non-red component with many edges.

###### Proof of Theorem 1.

Fix a $k$-coloring $\chi$ of the edges of $K_{n}$, and suppose that the
largest number of edges in a monochromatic connected component in this
coloring is $z\binom{n}{2}$. Without loss of generality, let red be the color
with the most edges, so there are at least $\frac{1}{k}\binom{n}{2}$ red edges
in our coloring. Let $\mathcal{C}_{1}=\\{R_{1},R_{2},\dots,R_{m}\\}$ be the
set of red components, with

$|V(R_{1})|\geq|V(R_{2})|\geq\cdots\geq|V(R_{m})|,\qquad\sum_{i=1}^{m}|V(R_{i})|=n.$
(3.1)

Let $x=\frac{1}{\binom{n}{2}}\sum_{i=1}^{m}|E(R_{i})|$, so $x\geq\frac{1}{k}$.
By assumption, $|E(R_{i})|\leq z\binom{n}{2}$ for $1\leq i\leq m$. As in the
proof of Corollary 4, applying Theorem 3 with $K_{n}$ as $G$ and its red color
class (i.e. the spanning subgraph formed by its red edges) as $H$ yields a red
component with at least $x^{2}\binom{n}{2}$ edges, so by assumption we have
$z\geq x^{2}\geq\frac{1}{k^{2}}$. Define $\delta=k-\frac{1}{\sqrt{z}}\geq 0$,
so $z=\frac{1}{(k-\delta)^{2}}$.

For any $j\in[1,m-1]$, let $G_{j}$ be the complete graph on
$\bigcup_{i=j+1}^{m}V(R_{i})$, with its coloring induced by $\chi$. Applying
Theorem 3 with $G_{j}$ as $G$ and the red color class of $G_{j}$ as $H$ yields
a red connected component $H^{\prime}=R_{i}$ with
$|E(H^{\prime})|\geq\frac{|E(H)|^{2}}{|E(G_{j})|}$. Since

$|E(H)|=\sum_{i=j+1}^{m}|E(R_{i})|=x\binom{n}{2}-\sum_{i=1}^{j}|E(R_{i})|\geq
x\binom{n}{2}-jz\binom{n}{2},$

while
$|E(G_{j})|=\binom{|V(G_{j})|}{2}=\binom{n-\sum_{i=1}^{j}|V(R_{i})|}{2}$, we
have

$z\binom{n}{2}\geq|E(H^{\prime})|\geq\frac{\max(x-jz,0)^{2}\binom{n}{2}^{2}}{\binom{n-\sum_{i=1}^{j}|V(R_{i})|}{2}}.$

Since $\frac{\max(x-jz,0)}{\sqrt{z}}\leq\frac{x}{\sqrt{z}}\leq 1$, this
implies

$\binom{n-\sum_{i=1}^{j}|V(R_{i})|}{2}\geq\frac{\max(x-jz,0)^{2}}{z}\binom{n}{2}\geq\binom{\frac{\max(x-jz,0)}{\sqrt{z}}n}{2}.$

Since $n-\sum_{i=1}^{j}|V(R_{i})|\geq 1$, and the function $f(X)=\binom{X}{2}$
is increasing for $X\geq 1$, this then implies

$\sum_{i=1}^{j}|V(R_{i})|\leq(1-x/\sqrt{z}+j\sqrt{z})n\qquad\text{for all
}j\in[1,m-1].$ (3.2)

We can now give an upper bound on $\sum_{i=1}^{m}\binom{|V(R_{i})|}{2}$ by
solving the corresponding convex optimization problem. The proof of the
following technical lemma will be deferred until the end of the section.

######  Lemma 7.

Let $x,z>0$. Subject to the constraints $v_{1}\geq\cdots\geq v_{m}\geq 0$,
$\sum_{i=1}^{m}v_{i}=1$, and

$\sum_{i=1}^{j}v_{i}\leq 1-x/\sqrt{z}+j\sqrt{z}\qquad\text{for all
}j\in[1,m-1],$

the quantity $\sum_{i=1}^{m}v_{i}^{2}$ is maximized when
$v_{1}=1-x/\sqrt{z}+\sqrt{z}$, $v_{i}=\sqrt{z}$ for $2\leq
i\leq\lfloor\frac{x}{z}\rfloor$,
$v_{\lfloor\frac{x}{z}\rfloor+1}=x/\sqrt{z}-\lfloor\frac{x}{z}\rfloor\sqrt{z}$,
and $v_{i}=0$ for $i>\lfloor\frac{x}{z}\rfloor+1$.

Let $v_{i}=\frac{|V(R_{i})|}{n}$. Since (3.1) and (3.2) hold, we can apply
Lemma 7 to obtain

$\sum_{i=1}^{m}v_{i}^{2}\leq(1-x/\sqrt{z}+\sqrt{z})^{2}+\left(\left\lfloor\frac{x}{z}\right\rfloor-1\right)z+\left(\left(x/z-\left\lfloor\frac{x}{z}\right\rfloor\right)\sqrt{z}\right)^{2}\leq(1-x/\sqrt{z}+\sqrt{z})^{2}+(x/z-1)z,$

so that we have

$\displaystyle\sum_{i=1}^{m}\binom{|V(R_{i})|}{2}$
$\displaystyle=\frac{n^{2}\sum_{i=1}^{m}v_{i}^{2}-n}{2}\leq\frac{n^{2}((1-x/\sqrt{z}+\sqrt{z})^{2}+(x/z-1)z)-n}{2}.$

Finally, let $G$ be the complete $m$-partite graph obtained from $K_{n}$ by
removing all edges within each $V(R_{i})$, with its coloring induced by
$\chi$. There are no red edges in $G$, so by the pigeonhole principle, one of
the $k-1$ remaining colors has at least $\frac{1}{k-1}|E(G)|$ edges in $G$.
Let $H$ be the spanning subgraph of $G$ induced by the edges in that color.
Applying Theorem 3 then yields a monochromatic connected component
$H^{\prime}$ with at least $\frac{1}{(k-1)^{2}}|E(G)|$ edges. Then by
assumption we have

$\displaystyle z$
$\displaystyle\geq\frac{1}{(k-1)^{2}}\frac{|E(G)|}{\binom{n}{2}}=\frac{1}{(k-1)^{2}}\left(1-\frac{\sum_{i=1}^{m}\binom{|V(R_{i})|}{2}}{\binom{n}{2}}\right)$
$\displaystyle\geq\frac{1}{(k-1)^{2}}\left(1-\frac{n^{2}((1-x/\sqrt{z}+\sqrt{z})^{2}+(x/z-1)z)-n}{n^{2}-n}\right)$
$\displaystyle\geq\frac{1}{(k-1)^{2}}\left(1-(1-x/\sqrt{z}+\sqrt{z})^{2}-(x/z-1)z\right),$

which rearranges to give

$1-(k-1)^{2}z\leq(1-x/\sqrt{z}+\sqrt{z})^{2}+(x/z-1)z=\frac{1}{z}x^{2}-(1+2/\sqrt{z})x+(1+2\sqrt{z}).$

The right hand side is a quadratic in $x$ that is decreasing for
$x\leq\frac{z}{2}+\sqrt{z}$. Since by assumption $\sqrt{z}\geq
x\geq\frac{1}{k}$, we then have

$1-(k-1)^{2}z\leq\frac{1}{zk^{2}}-(1+2/\sqrt{z})\frac{1}{k}+(1+2\sqrt{z}).$

Substituting in $z=\frac{1}{(k-\delta)^{2}}$ yields

$1-\frac{(k-1)^{2}}{(k-\delta)^{2}}\leq\frac{(k-\delta)^{2}}{k^{2}}-\frac{1+2(k-\delta)}{k}+1+\frac{2}{k-\delta},$

which upon rearrangement becomes

$\displaystyle 0$
$\displaystyle\leq(k-1)^{2}k^{2}+(k-\delta)^{4}-k(k-\delta)^{2}-2(k-\delta)^{3}k+2(k-\delta)k^{2}$
$\displaystyle=-k^{3}+k^{2}-k\delta^{2}+2k^{3}\delta-2k\delta^{3}+\delta^{4}$
$\displaystyle=(k-\delta^{2})^{2}-k(k^{2}-\delta^{2})(1-2\delta).$

This implies

$1-2\delta\leq\frac{(k-\delta^{2})^{2}}{k(k^{2}-\delta^{2})}<\frac{1}{k},$

so that $\delta>\frac{k-1}{2k}$. Thus, the coloring $\chi$ contains a
monochromatic connected component with at least $z\binom{n}{2}$ edges, where

$z=\frac{1}{(k-\delta)^{2}}>\frac{1}{(k-\frac{1}{2}+\frac{1}{2k})^{2}}\geq\frac{1}{k^{2}-k+\frac{1}{4}+1-\frac{1}{4k}+\frac{1}{4k^{2}}}\geq\frac{1}{k^{2}-k+\frac{5}{4}},$

as desired. ∎

###### Proof of Lemma 7.

Since the feasible region is compact, there exists a choice of the $v_{i}$
such that the desired maximum is attained. Fix such a maximizing choice of the
$v_{i}$. For $j\in[m]$, let $A_{j}$ denote the given constraint on
$\sum_{i=1}^{j}v_{i}$, and let $S$ be the set of $j$ for which equality holds
in $A_{j}$. Let $m^{\prime}=\lfloor\frac{x}{z}\rfloor$. Since
$\sum_{i=1}^{m}v_{i}=1$, the constraints $A_{j}$ for $j>m^{\prime}$ cannot be
tight, so $S\subseteq[m^{\prime}]$. For any $i<j$ and any
$\varepsilon\in(0,v_{j}]$, replacing $(v_{i},v_{j})$ with
$(v_{i}+\varepsilon,v_{j}-\varepsilon)$ increases the value of
$\sum_{i=1}^{m}v_{i}^{2}$, so by maximality, no such “smoothing” operation is
possible. That is, we can assume the following equality condition for all
$(i,j)$ with $i<j$: Either $v_{i}=v_{i-1}$, or $v_{j}=v_{j+1}$, or
$S\cap[i,j-1]\neq\emptyset$. If $S=[m^{\prime}]$, then we are in exactly the
maximizing case described (since by the equality conditions for
$(m^{\prime}+1,i)$ we have $v_{i}=0$ for all $i\geq m^{\prime}+2$), so assume
otherwise.

First, suppose there is some $i_{0}\in[m^{\prime}]$ such that
$v_{i_{0}}<\sqrt{z}$, and pick the smallest such index $i_{0}$. Let
$i_{1}\in[m]$ be the largest index such that $v_{i_{1}}>0$, and note that
$i_{1}>i_{0}$ since $\sum_{i=1}^{i_{0}}v_{i}<1$. Then we have
$v_{i_{0}}<v_{i_{0}-1}$ and $v_{i_{1}}>v_{i_{1}+1}$. But as
$\sum_{i=1}^{j}v_{i}\leq(1-x/\sqrt{z}+(j-1)\sqrt{z})+v_{j}<(1-x/\sqrt{z}+(j-1)\sqrt{z})+\sqrt{z}$
for any $j\geq i_{0}$, we have $S\cap[i_{0},i_{1}]=\emptyset$, contradicting
the equality condition for $(i_{0},i_{1})$. So, we can assume
$v_{i}\geq\sqrt{z}$ for all $i\leq m^{\prime}$. In particular, if $j\in S$ for
some $j\in[m^{\prime}]$, then we recursively obtain $v_{i}=\sqrt{z}$ for all
$i\in[j+1,m^{\prime}]$, so $[j,m^{\prime}]\subseteq S$.

Thus, we can assume $1\notin S$, i.e. $v_{1}<1-x/\sqrt{z}+\sqrt{z}$. By the
equality condition for $(1,2)$, we must then have $v_{2}=v_{3}$. Let
$j_{0}\geq 3$ be the smallest index such that $v_{j_{0}+1}\neq v_{j_{0}}$. By
the equality condition for $(1,j_{0})$, there is some $j_{1}\in
S\cap[2,j_{0}-1]$. Then

$1-x/\sqrt{z}+j_{1}\sqrt{z}=\sum_{i=1}^{j_{1}}v_{i}=v_{1}+(j_{1}-1)v_{2}<1-x/\sqrt{z}+\sqrt{z}+(j_{1}-1)v_{2},$

which implies $v_{2}>\sqrt{z}$. But then
$\sum_{i=1}^{j_{1}+1}v_{i}=1-x/\sqrt{z}+\sqrt{z}+j_{1}v_{2}>1-x/\sqrt{z}+(j_{1}+1)\sqrt{z}$,
violating $A_{j_{1}+1}$. This is a contradiction, so in fact $S=[m^{\prime}]$,
and we are in the desired maximizing case. ∎

We remark that in Gyárfás’s construction, after removing all edges contained
in the vertex set of each red component, each non-red component is left with
approximately
$\frac{1}{(k-1)^{2}}(1-\frac{1}{k-1})\binom{n}{2}=\frac{k-2}{(k-1)^{3}}\binom{n}{2}$
edges. Since $\frac{k-2}{(k-1)^{3}}=\frac{1}{k^{2}-k+1+\frac{1}{k-2}}$, this
suggests that the method used to prove Theorem 1 cannot show a lower bound on
$M(n,k)$ better than $\frac{1}{k^{2}-k+1}\binom{n}{2}$ without introducing
additional ideas.

###  3.2. Multicolor Ramsey numbers of trails and circuits

A _trail_ is a walk without repeated edges, and a _circuit_ is a trail with
the same first and last vertex. The ($k$-color) Ramsey problem for trails is
the question of finding the largest $m$ such that every $k$-coloring of the
edges of $K_{n}$ contains a monochromatic trail of length $m$.

Answering a question of Osumi [6], Conlon and Tyomkyn [1] show that every
$2$-coloring of the edges of $K_{n}$ contains a monochromatic circuit with at
least $\frac{2}{9}n^{2}+O(n^{3/2})$ edges, and this is asymptotically tight.
For the case of general $k$, they observe that by deleting a forest in each
color class of a $k$-coloring of $K_{n}$ to make each color class Eulerian
(i.e. ensuring every vertex has even degree in each color), one can reduce
this Ramsey problem to a variant of the problem of determining $M(n,k)$. Where
previously we colored the edges of $K_{n}$ and found a large monochromatic
component, we now apply the same procedure to the graph obtained by deleting
at most $kn$ edges from $K_{n}$. We now prove a lower bound for the general
case of this problem, analogous to the bound on $M(n,k)$ given in Theorem 1.

######  Corollary 8.

Every $k$-coloring of the edges of $K_{n}$ contains a monochromatic circuit
(and hence a monochromatic trail) of length at least
$\frac{1}{k^{2}-k+\frac{5}{4}}\binom{n}{2}+O_{k}(n)$.

###### Proof.

Fix a $k$-coloring of $E(K_{n})$. As in [1], we can remove from each color
class a forest that meets all odd degree vertices, leaving a coloring where
every color class, and hence every monochromatic connected component, is
Eulerian. Let $\chi$ be the resulting partial $k$-coloring of $E(K_{n})$,
where the (at most $kn$) removed edges are left uncolored, and let
$z\binom{n}{2}$ be the largest number of edges in a monochromatic component in
this coloring. It suffices to show that
$z\geq\frac{1}{k^{2}-k+\frac{5}{4}}+O_{k}(n^{-1})$, since every monochromatic
component is Eulerian, and thus contains an Eulerian circuit. Fix a color
(say, red) with $x\binom{n}{2}$ edges, where
$x\geq\frac{1}{k}\frac{\binom{n}{2}-nk}{\binom{n}{2}}=\frac{1}{k}-\frac{2}{n-1}$.
As before, applying Theorem 3 with $G=K_{n}$ immediately yields $z\geq
x^{2}\geq\frac{1}{k^{2}}-O(n^{-1})$. Note that this means there is a
monochromatic circuit with at least
$\left(\frac{1}{k^{2}}-O(n^{-1})\right)\binom{n}{2}=\frac{1}{k^{2}}\binom{n}{2}+O(n)$
edges.

To improve this lower bound further, we proceed as in the proof of Theorem 1.
Letting $R_{1},\dots,R_{m}$ be the red components, such that (3.1) holds, we
obtain (3.2) in the same manner as before, so we can once again apply Lemma 7
to obtain the same upper bound on $\sum_{i=1}^{m}\binom{|V(R_{i})|}{2}$ in
terms of $x$ and $z$. Let $G$ be the complete $m$-partite graph with parts
$V(R_{1}),\dots,V(R_{m})$, with its partial coloring induced by $\chi$. Since
$G$ has no red edges, and at most $kn$ edges are uncolored, one of its color
classes $H$ has at least
$\frac{1}{k-1}(|E(G)|-kn)=\left(\frac{1}{k-1}-O_{k}(n^{-1})\right)|E(G)|$
edges. Applying Theorem 3 as before yields

$\displaystyle z$
$\displaystyle\geq\left(\frac{1}{k-1}-O_{k}(n^{-1})\right)^{2}\frac{|E(G)|}{\binom{n}{2}}\geq\left(\frac{1}{(k-1)^{2}}-O_{k}(n^{-1})\right)\left(1-\frac{\sum_{i=1}^{m}\binom{|V(R_{i})|}{2}}{\binom{n}{2}}\right)$
$\displaystyle\geq\left(\frac{1}{(k-1)^{2}+O_{k}(n^{-1})}\right)\left(1-(1-x/\sqrt{z}+\sqrt{z})^{2}-(x/z-1)z\right).$

Letting $z=\frac{1}{(k-\delta)^{2}}$ and noting that
$x\geq\frac{1}{k}-O(n^{-1})$, we can perform the same rearrangements and
substitutions as in the proof of Theorem 1, simply separating out the
$O_{k}(n^{-1})$ terms at each step, to derive the inequality
$1-2\delta<\frac{1}{k}+O_{k}(n^{-1})$, and thus

$z\geq\frac{1}{k^{2}-k+\frac{5}{4}-O_{k}(n^{-1})}=\frac{1}{k^{2}-k+\frac{5}{4}}+O_{k}(n^{-1}).$

Then the coloring $\chi$ contains a monochromatic connected component, and
hence a monochromatic Eulerian circuit, with at least
$z\binom{n}{2}\geq\frac{1}{k^{2}-k+\frac{5}{4}}\binom{n}{2}+O_{k}(n)$ edges,
as claimed. ∎

###  3.3. Three colors

In this section, we prove the lower and upper bounds on $M(n,3)$ in separate
lemmas in order to establish Theorem 2. We then discuss the behavior of
$M(n,3)$ for small values of $n$, and conclude by describing how to adapt our
proofs to obtain asymptotically tight bounds for the Ramsey numbers of trails
and circuits in three colors.

######  Lemma 9.

Every $3$-coloring of the edges of $K_{n}$ contains a monochromatic component
with at least $\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges.

###### Proof.

Let $G=K_{n}$, and call the three colors red, green, and blue. First, suppose
one of the color classes (say, red) is connected. If there are at least
$\frac{1}{6}|E(G)|$ red edges, we are done. Otherwise, the other two colors
together have at least $\frac{5}{6}|E(G)|$ edges, so one of them has at least
$\frac{5}{12}|E(G)|$ edges. Applying Theorem 3 with $G=K_{n}$ to the graph in
that color then gives a monochromatic component with at least
$(\frac{5}{12})^{2}|E(G)|>\frac{1}{6}|E(G)|$ edges, so we are again done.

Thus, we can assume every color has at least two components. Without loss of
generality, let red be the color with the most edges, so the red graph $H$ has
at least $\frac{1}{3}|E(G)|$ edges. Let $H^{\prime}$ be the component of $H$
with the highest average degree, so
$|V(H^{\prime})|\geq\bar{d}(H^{\prime})+1\geq\frac{1}{3}n$. We can assume
$|V(H^{\prime})|\leq\frac{1}{2}n$; otherwise, we would have
$|E(H^{\prime})|\geq\frac{1}{2}|V(H^{\prime})|\bar{d}(H^{\prime})>\frac{1}{6}|E(G)|$.
Let $V_{1}=V(H^{\prime})$ and $V_{2}=V(G)\setminus V_{1}$, and let
$G^{\prime}$ be the bipartite graph induced by $G$ between $V_{1}$ and
$V_{2}$, so $G^{\prime}$ has at least
$|V_{1}||V_{2}|\geq\frac{2}{9}n^{2}>\frac{1}{3}|E(G)|$ edges, all of which
must be green or blue.

Fix an edge in $G^{\prime}$ and consider the monochromatic component $C_{1}$
of $G^{\prime}$ containing this edge. Without loss of generality, assume
$C_{1}$ is green. Suppose $C_{1}$ covers all of $V_{1}$. Since every green
edge in $G^{\prime}$ intersects $V_{1}$, this means there is exactly one green
component of $G^{\prime}$ with a nonzero number of edges. Since there are at
least two green components in $G$, there is a vertex $v\in V_{2}$ not in
$C_{1}$. Then all edges between $v$ and $V_{1}$ must be blue, so all vertices
of $V_{1}$ are in the same blue component in $G^{\prime}$. This implies that
there is also exactly one blue component of $G^{\prime}$ with nonzeroly many
edges. Thus, all edges of $G^{\prime}$ are in one of two monochromatic
components, one of which then has at least
$\frac{1}{2}|E(G^{\prime})|>\frac{1}{6}|E(G)|$ edges, as desired.

Figure 1. Lower bound: Exactly two components of each color

We are likewise done if $C_{1}$ covers all of $V_{2}$, so we can assume
$V_{1}\setminus C_{1}$ and $V_{2}\setminus C_{1}$ are both nonempty. Let
$A_{1}=V_{1}\cap C_{1}$, $B_{1}=V_{2}\cap C_{1}$, $A_{2}=V_{1}\setminus
A_{1}$, $B_{2}=V_{2}\setminus B_{1}$. Then all edges between $A_{1}$ and
$B_{2}$, or between $A_{2}$ and $B_{1}$, can only be blue. Then there are at
most two blue components, and thus by assumption exactly two. This in turn
implies that all edges between $A_{1}$ and $B_{1}$, or between $A_{2}$ and
$B_{2}$, can only be green, so there are exactly two green components $C_{1}$
and $C_{2}$. Finally, all edges between $B_{1}$ and $B_{2}$ can only be red,
so we conclude that there are exactly two red components, $V_{1}$ and $V_{2}$;
see Figure 1.

Since each of the three colors has exactly two components, one of the
components has at least $\frac{1}{6}|E(G)|$ edges, as desired. ∎

######  Lemma 10.

For sufficiently large $n$, there exists a $3$-coloring of the edges of
$K_{n}$ such that every monochromatic component contains at most
$\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges.

###### Proof.

Figure 2. Initial construction for the $M(n,3)$ upper bound

We consider the following modification of a construction by Gyárfás: Let
$V=V_{1}\cup V_{2}\cup V_{3}\cup V_{4}$ be a partition of the vertices of
$G=K_{n}$ into four parts, with

$\left\lceil\frac{n}{4}\right\rceil=|V_{1}|\geq\cdots\geq|V_{4}|=\left\lfloor\frac{n}{4}\right\rfloor.$

Letting $E(U,W)$ denote the set of edges between vertex sets $U$ and $W$,
color $E(V_{1},V_{2})\cup E(V_{3},V_{4})$ red, $E(V_{1},V_{3})\cup
E(V_{2},V_{4})$ green, and $E(V_{1},V_{4})\cup E(V_{2},V_{3})$ blue. For large
enough $n$, $e(V_{i},V_{j})\leq\lceil\frac{1}{6}\binom{n}{2}\rceil$ for $1\leq
i<j\leq 4$, so it remains to extend this partial coloring by coloring the
edges within each of the $V_{i}$ such that each monochromatic component has at
most $\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges at the end. It is natural to
attempt the simple approach of distributing the edges within each $V_{i}$ as
evenly as possible among the three colors. However, the possibility of a
slight difference in size among the $V_{i}$ can yield a $\Theta(n)$ difference
among the numbers of edges in each component if we are not sufficiently
careful; a different coloring strategy will turn out to be simpler to analyze.

We call an extension of the above partial coloring _nice_ if each green or
blue component contains exactly $\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges.
At least one nice coloring exists: we can color exactly
$\lceil\frac{1}{6}\binom{n}{2}\rceil-e(V_{1},V_{3})$ of the edges within
$V_{1}$ and $\lceil\frac{1}{6}\binom{n}{2}\rceil-e(V_{2},V_{4})$ of the edges
within $V_{2}$ green, and exactly
$\lceil\frac{1}{6}\binom{n}{2}\rceil-e(V_{2},V_{3})$ of the edges within
$V_{3}$ and $\lceil\frac{1}{6}\binom{n}{2}\rceil-e(V_{1},V_{4})$ of the edges
within $V_{4}$ blue (there are enough edges within each $V_{i}$ to do this
when $n$ is sufficiently large), and color all remaining edges red. See Figure
2 for a diagram of this coloring. Fix a nice coloring where the larger of the
two red components contains as few edges as possible.

Suppose one of the red components in this coloring, without loss of generality
the one on $V_{1}\cup V_{2}$, has more than
$\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges. Then $V_{3}\cup V_{4}$ must have
less than $\lceil\frac{1}{6}\binom{n}{2}\rceil$ red edges. Without loss of
generality let $V_{1}$ contain a red edge $e$. If either $V_{3}$ contains a
green edge, or $V_{4}$ contains a blue edge, we can switch the color of that
edge with edge $e$, preserving the sizes of the green and blue components
while decreasing the size of the larger red component by one. Otherwise,
$V_{3}$ is entirely blue and red, and $V_{4}$ is entirely green and red. Since
the red component on $V_{3}\cup V_{4}$ has less than
$\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges, neither $V_{3}$ nor $V_{4}$ can
be entirely red (for sufficiently large $n$,
$\lfloor\frac{n}{4}\rfloor^{2}+\binom{\lfloor\frac{n}{4}\rfloor}{2}>\lceil\frac{1}{6}\binom{n}{2}\rceil$).
Then if $V_{2}$ contains a red edge, we can similarly switch two edges to
reduce the size of the larger red component by one, so we can assume $V_{2}$
is entirely green and blue. But then there is one component of each color
(including the larger red component) that does not have any edges within
$V_{2}\cup V_{3}\cup V_{4}$, which means there are at least
$3\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges incident to $V_{1}$, a
contradiction since

$e(V_{1},V)\leq\binom{\lceil\frac{n}{4}\rceil}{2}+\left\lceil\frac{n}{4}\right\rceil\left(n-\left\lceil\frac{n}{4}\right\rceil\right)<3\left\lceil\frac{1}{6}\binom{n}{2}\right\rceil,$

for sufficiently large $n$. Thus indeed there is a construction of this form
where every monochromatic component has at most
$\lceil\frac{1}{6}\binom{n}{2}\rceil$ edges. ∎

Combining Lemmas 9 and 10 yields Theorem 2 as desired.

The construction in the proof of Lemma 10 is well-defined for all $n\geq 46$.
A more careful analysis, splitting into cases based on the value of $n$ modulo
$4$ and then using a more explicit construction in each case, shows that in
fact $M(n,3)=\left\lceil\frac{1}{6}\binom{n}{2}\right\rceil$ for all $n\geq
11$ except $n=13,17$. However, the lower bound from Lemma 9 is not sharp for
some of these small values of $n$. Closely inspecting each step of our proof
for $n=17$, for example, we can deduce that $M(17,3)=24$, instead of the
expected $23$; indeed, all of the bounds before the final step in the proof of
Lemma 9 are loose enough to yield a component with at least $24$ edges unless
we are in the case depicted in Figure 1, where one of the vertex sets $A_{i}$
or $B_{i}$ has size $5$, and the other three sets have size $4$. The set of
size $5$ then contains $10$ internal edges, some $4$ of which are in the same
color, yielding a component with $4+20=24$ edges as claimed. This shows a
genuine difference in the behavior of $M(n,3)$ for these small values of $n$
due to integer-related constraints in the extremal configurations.

We can adjust the proof of Lemma 9 to give a lower bound on the size of the
largest monochromatic circuit in a $3$-coloring of the edges of $K_{n}$, as
follows. First, as before, we can remove a forest in each color and leave each
color class Eulerian. The resulting graph $G$ has at least $\binom{n}{2}-3n$
edges, so all but at most $3\sqrt{n}$ of the vertices have degree at least
$n-1-2\sqrt{n}$. We then pass to the induced subgraph $G^{\prime}$ on these
$n^{\prime}\geq n-3\sqrt{n}$ vertices; the minimum degree of $G^{\prime}$ is
at least $n^{\prime}-1-2\sqrt{n}\geq n^{\prime}-3\sqrt{n^{\prime}}$ when $n$
is sufficiently large. The argument then proceeds largely as in the proof of
Lemma 9, except that the condition of every green or blue component
intersecting both $V_{1}$ and $V_{2}$ is strengthened by requiring every green
or blue component to intersect each of $V_{1}$ and $V_{2}$ in more than
$6\sqrt{n^{\prime}}$ vertices. The proof then concludes as before, reducing to
the case where there are exactly two components in each color. When the edges
between $V(G^{\prime})$ and $V(G)\setminus V(G^{\prime})$ are added back in,
it remains true that there are at most two components in each color with a
positive number of edges. So, at least one monochromatic component in $G$ has
at least $\frac{1}{6}(|E(G)|-(3\sqrt{n})^{2})\geq\frac{1}{12}n^{2}-O(n)$
edges. Since this component of $G$ is Eulerian, we have a circuit, and hence a
trail, of the desired length, for all sufficiently large $n$. The upper bound
from Lemma 10 likewise applies to the size of the longest circuit, showing
that the lower bound is asymptotically tight.

Acknowledgments. I would like to thank my advisor Jacob Fox for introducing me
to this problem and for helpful conversations along the way, as well as David
Conlon and Mykhaylo Tyomkyn for the insights from our later joint work that
served as inspiration for some of the strengthened arguments in the revised
version of this paper. In addition, I would like to thank the anonymous
referees for their careful reading and helpful comments, including a specific
suggestion that led to a significant strengthening in the bound for general
$k$ in Theorem 1.

## References

  * [1] Conlon, D., and Tyomkyn, M. Ramsey numbers of trails and circuits.  arXiv preprint arXiv:2109.02633 (2021).
  * [2] DeBiasio, L., Krueger, R. A., and Sárközy, G. N. Large monochromatic components in multicolored bipartite graphs.  J. Graph Theory 94, 1 (2020), 117–130.
  * [3] Gyárfás, A. Partition coverings and blocking sets in hypergraphs.  Communications of the Computer and Automation Institute of the Hungarian Academy of Sciences 71 (1977), 62.
  * [4] Gyárfás, A. Large monochromatic components in edge colorings of graphs: a survey.  In Ramsey theory, vol. 285 of Progr. Math. Birkhäuser/Springer, New York, 2011, pp. 77–96.
  * [5] Gyárfás, A., and Sárközy, G. N. Large monochromatic components in edge colored graphs with a minimum degree condition.  Electron. J. Combin. 24, 3 (2017), Paper No. 3.54, 14.
  * [6] Osumi, M. Ramsey numbers of trails.  arXiv preprint arXiv:2109.00734 (2021).
  * [7] Ruszinkó, M. Large components in $r$-edge-colorings of $K_{n}$ have diameter at most five.  J. Graph Theory 69, 3 (2012), 337–340.


(115)

In other words, spin configuration is along $\pm\mathbf{\hat{z}}$, and Fermi
surface distortion is along the $x$-axis. Three collective modes are Goldstone
modes, including one branch in the density channel, and two branches in the
spin channel.

The density channel Goldstone mode is the oscillation of the distorted Fermi
surface. It is associated with the field $n_{2}^{z}$,

$\displaystyle
n^{z}_{2}(\mathbf{q})=-\frac{f^{a}_{1}(\mathbf{q})}{V}\sum_{\mathbf{k}}\psi^{\dagger}_{\mathbf{k+q}}\vec{\sigma}\psi_{\mathbf{k}}k_{y},$
(116)

which describes the Fermi surface oscillation in the $y$-direction while
keeping the spin configuration unchanged. Calculations at the random-phase
approximation (RPA) level show the effective Lagrangian,

$\displaystyle
L_{FS}^{\alpha}(\mathbf{q},\omega)=N_{0}\left[\frac{q\xi^{2}}{|F^{a}_{l}|}-i\frac{\omega}{2v_{f}q}(1+2\cos
2\phi_{q})\right].$ (117)

This Goldstone mode is overdamped because of the Landau damping, and the
damping is anisotropic depending on propagation directions.

The spin channel Goldstone modes $n_{sp,x\pm iy}$ describe spin oscillations
while keeping the Fermi surface unchanged, which are spin-dipole precession
modes. In contrast, they exhibit nearly isotropic underdamped dispersion
relations at small propagating wavevectors with the dispersion relation

$\displaystyle\omega^{2}_{x\pm iy}=\frac{\bar{n}^{2}}{|F^{a}_{1}|}(q\xi)^{2}.$
(118)

Different from spin-waves in the ferromagnets, the dispersion relation here is
linear with momentum, which is a consequence of time-reversal symmetry.

####  IV.4.2 Goldstone modes in the $\beta$-phase

We further study the Goldstone modes in the $\beta$-phase. For simplicity, we
consider the 3D $\beta$-phase with the isotropic ground state exhibiting

$\displaystyle n^{\mu a}=\bar{n}\delta_{\mu a}.$ (119)

In other words, $\mathbf{n_{1,2,3}}$ form an orthogonal triad. The total
angular momentum $\mathbf{J}$ remains conserved, such that fluctuations of
$\delta n^{\mu a}$ are classified into eigenstates of $\mathbf{J}$ as
$O_{jj_{z}}(\mathbf{q},\omega)$. $j=0,1,2$ mean the singlet, triplet and
quintet channels respectively, and $j_{z}$ is the SO(2) quantum number
rotating around the propagation direction $\mathbf{q}$.

The Goldstone modes belong to the triplet channel $(j=1)$, which are the small
relative spin-orbit rotations as

$\displaystyle O_{1,0}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\epsilon_{z\mu a}\delta n^{\mu
a}(\mathbf{q},\omega),\ \ \ $ $\displaystyle O_{1,\pm 1}(\mathbf{q},\omega)$
$\displaystyle=$ $\displaystyle\frac{1}{2}(\epsilon_{x\mu a}\pm
i\epsilon_{y\mu a})\delta n^{\mu a}(\mathbf{q},\omega).$ (120)

The RPA approximation gives the dispersion relations,

$\displaystyle\omega^{2}=4\bar{n}^{2}|F^{a}_{1}|\Big{(}\frac{\kappa
q^{2}}{N_{f}}+\frac{j_{z}|q|x}{18k_{f}}\Big{)}\ \ \
(x=\frac{\bar{n}}{v_{f}k_{f}},j_{z}=0,\pm 1),$

which is valid in the low energy regime of $\omega,v_{f}q\ll\bar{n}\ll
v_{f}k_{f}$.

The linear dependence on $q$ in the dispersion Eq. (IV.4.2) is due to the
broken parity in the ordered $\beta$ phase. Consequently, $\omega^{2}$ becomes
negative for the branch with helicity $j_{z}=-1$ at small $q$, This means that
the uniform ground state in the $\beta$-phase is unstable, instead it exhibits
a Lifshitz-like instability. This behavior is a general feature in systems
with broken parity such as the spiral order in helical magnets, and the
cholesteric liquid crystalsGennes and Prost (1995). The true ground state
configuration in the $\beta$ phase is complicated with the preliminary
analysis presented in Ref. Wu  _et al._ (2007).

####  IV.4.3 Resonances in inelastic neutron scattering spectroscopy

The unconventional magnetic orders are spin-multipole moment in momentum space
and cannot couple to neutron magnetic moments statically. Hence, there should
be no elastic Bragg peaks. The spin-channel Goldstone modes in the
$\alpha$-phase do not couple to neutron moments directly, either.
Nevertheless, they carry spin quantum numbers and thus couple to spin-wave
modes dynamically.

Consider the following commutation relations,

$\displaystyle[S_{x},n^{y}_{1}]=in^{z}_{1},\ \
\,[S_{y},n^{x}_{1}]=-in^{z}_{1}.$ (122)

In the $p$-wave $\alpha$-phase with the configuration given in Eq. (115),
$n^{z}_{1}$ can be replaced by the constants of $\pm i\bar{n}$. As a result,
the Goldstone modes $n_{1}^{x}$ and $n_{1}^{y}$ become conjugate to spin, and
the coupling to spin is developed dynamically.

More formally, we can write down the following coupling Lagrangian,

$\displaystyle
L=\left(\vec{n}_{1}\times\partial_{t}\vec{n}_{1}+\vec{n}_{2}\times\partial_{t}\vec{n}_{2}\right)\cdot\vec{S}.$
(123)

In the ordered state of Eq. (115), it is reduced to

$\displaystyle
L=\bar{n}\left(S_{y}\partial_{t}n_{1x}-S_{x}\partial_{t}n_{1y}\right).$ (124)

The RPA approximation shows that the dynamic spin-spin correlation function
behaves as

$\displaystyle\chi_{s}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle\langle
S_{+}(\mathbf{q},\omega)S_{-}(-\mathbf{q},-\omega)\rangle$ (125)
$\displaystyle=$
$\displaystyle\frac{N_{0}\frac{\omega^{2}}{\bar{n}^{2}}}{\frac{\kappa
q^{2}}{N_{0}}-\frac{2}{|F^{a}_{1}|}\frac{\omega^{2}}{\bar{n}^{2}}-i\delta}.$

Hence, it induces a resonance part in the transverse spin wave-excitations.
The spectral functions exhibit the $\delta$-peak at the excitation energy of
the Goldstone mode,

$\displaystyle\mbox{Im}\chi_{s}(\mathbf{q},\omega)=\kappa\pi
v_{f}^{2}\bar{n}^{2}q^{2}|F_{1}^{a}|^{2}\delta(\omega^{2}-\omega_{q}^{2}),$
(126)

which can be detected in the inelastic neutron scattering experiments. This is
very similar to the interpretation of the SO(5) theory to the neutron
resonance mode: the $\pi$-mode lies in the particle-particle channel which
decouples from spin in the normal state, but becomes conjugate to spin in the
superconducting state giving rise to spin resonances Demler and Zhang (1995);
Demler  _et al._ (2004); Zhang (1997).

Such a resonance peak only exhibits in the ordered phase, and vanishes in the
disordered phase. As shown in Eq. (124), in the anisotropic $\alpha$-phases
this resonance only occurs in spin-flip channels. Similar analysis can also be
performed in the isotropic $\beta$-phases, in which the resonances occur in
both spin-flip and non-flip channels Wu  _et al._ (2007).

###  IV.5 Spin-orbit coupled Fermi liquid theory

So far we have considered the dynamic generation of spin-orbit coupling in
non-relativistic Fermi liquid theory. Nevertheless in materials with heavy
elements, there does exist the relativistic spin-orbit coupling.

If a system does not exhibit inversion symmetry, the relativistic spin-orbit
coupling leads to Fermi surface splitting, say, the Rashba type. In this case,
the relativistic spin-orbit coupling behaves like an external field which
would round off the unconventional phase transition and pin down a particular
spin-orbit ordering configuration. This situation is similar to cooling a
magnet below the transition temperature in an external magnetic field.

On the other hand, if a system Hamiltonian still preserves both parity and
time-reversal symmetries, the Fermi surface should remain doubly degenerate.
Spin-orbit coupling does not manifest itself in the Fermi surface splitting
but should exhibit in the Landau Fermi liquid theory. Such a situation also
occurs in the presence of prominent magnetic dipolar interactions, which is
invariant under simultaneous rotations in both orbital and spin channels, but
not under a rotation in either channel. Landau-Fermi liquid theory has been
extended to this situation Li and Wu (2012); Fu (2015).

In the inversion invariant spin-orbit coupled systems, the fermion
distribution function is reorganized in the spin-orbit coupled bases as

$\displaystyle\delta n_{\alpha\alpha^{\prime}}(\hat{k})$ $\displaystyle=$
$\displaystyle\sum_{JJ_{z};LS}\delta
n_{JJ_{z};LS}~{}Y_{JJ_{z};LS}(\hat{k},\alpha\alpha^{\prime}),$ (127)

where $Y_{JJz;LS}(\hat{k},\alpha\alpha^{\prime})$ is the spin-orbit coupled
spherical harmonic functions

$\displaystyle Y_{JJz;LS}(\hat{k},\alpha\alpha^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{ms_{z}}\langle LmSs_{z}|JJ_{z}\rangle
Y_{Lm}(\hat{k})\chi_{Ss_{z},\alpha\alpha^{\prime}},$

and $\chi_{Ss_{z},\alpha\alpha^{\prime}}$ is the bases for the particle-hole
singlet (density) channel and triplet (spin) channel, respectively. The Landau
interaction function is generalized to the interaction matrix,

$\displaystyle\frac{N_{0}}{4\pi}f_{\alpha\alpha^{\prime};\beta\beta^{\prime}}(\hat{k},\hat{k}^{\prime})$
$\displaystyle=$
$\displaystyle\sum_{JJ_{z}LL^{\prime}}Y_{JJz;L1}(\hat{k},\alpha\alpha^{\prime})F_{JJ_{z}L1;JJ_{z}L^{\prime}1}$
(129) $\displaystyle\times$ $\displaystyle
Y^{\dagger}_{JJz;L^{\prime}1}(\hat{k},\beta\beta^{\prime}),$

where we only keep the particle-hole triplet component. The Landau matrix is
diagonal with respect to the total angular momentum $J$ and its $z$-component
$J_{z}$, but may have off-diagonal elements with $L\neq L^{\prime}$.
Constrained by the inversion symmetry, $L-L^{\prime}=0,2$.

Similarly to the non-relativistic case, when an eigenvalue of the Landau
interaction matrix is negatively large, i.e, $\lambda<-1$, it triggers the
Pomeranchuk instability in the corresponding channel. For example, the
instability in the channel with $J=1^{-}$, $L=S=1$, where $``-"$ means odd
parity, generates the 3D analogy of the Rashba spin-orbit coupling,

$\displaystyle
H_{so,1^{-}}=|n|\sum_{k}\psi^{\dagger}(\mathbf{k})(\mathbf{k}\times\vec{\sigma})\cdot\mathbf{\hat{l}}\psi(\mathbf{k}),$
(130)

where $\mathbf{\hat{l}}$ is a 3D unit direction, $|n|$ is the magnitude of the
spin-orbit order parameter. The Pomeranchuk instability promotes it to the
single particle level by breaking the rotational symmetry and parity.

Let us still use the order parameter $n^{\mu b}$ defined in Sect. IV.2 to
represent the order parameters in the sector $L=S=1$ for a 3D inversion
invariant spin-orbit coupled Fermi liquid theory. The $3\times 3$ matrix of
$n$ includes three sectors of $J=0,1,2$, which corresponds to pseudo-scalar
(gryotropic), vector (Rashba), and tensor (Dresselhaus) type spin-orbit
coupling, respectively.

The Ginzburg-Landau free energy can be constructed as $F=F_{0}+\Delta F$,

$\displaystyle F_{0}$ $\displaystyle=$ $\displaystyle
r_{0}\mbox{tr}(n^{T}n)+\beta_{1}\left(\mbox{tr}(n^{T}n)\right)^{2}+\beta_{2}\mbox{tr}\left(n^{T}n\right)^{2},$
$\displaystyle\Delta F$ $\displaystyle=$
$\displaystyle\frac{r_{1}}{3}(\mbox{tr}n)^{2}+\frac{r_{2}}{4}\mbox{tr}(n^{T}-n)^{2}.$
(131)

Under $SO_{L}(3)$ and $SO_{S}(3)$ rotations, $n$ is transformed as $n\to
T_{S}nT_{L}^{\dagger}$, where $T_{L,S}$ is the rotation matrix in the orbit
and spin channels, respectively. $F_{0}$ is invariant under independent
$T_{L}$ and $T_{S}$, and $\Delta F$ is invariant under simultaneous spin-orbit
rotations, but not under a rotation in either channel. The $r_{1,2}$ terms are
an analogy to magnetic anisotropy for magnetic phase transitions, which lead
to different types of universal classes. Then at the quadratic level, the
eigenvalues of the pseudo-scalar, vector, tensor channels are determined by
$r_{0}+r_{1}$, $r_{0}+r_{2}$, and $r_{0}$, respectively. The actual ordering
depends on which eigenvalue is negatively most dominant. If the pseudo-scalar
channel instability dominates, the phase transition only breaks parity, which
is an Ising type transition without the Goldstone mode. If the vector channel
instability dominates, there is also rotational symmetry breaking with the
Goldstone manifold $S^{2}$. The symmetry breaking pattern for the tensor
ordering channel is more involved, the Goldstone manifold is formally denoted
as $SO(3)/G$, where $G$ is the residual symmetry group in the ordered state.
The nature of $G$ depends on in which tensor component the symmetry breaking
takes place. Nevertheless, when $\beta_{2}$ is included, the situation is
complicated, and the analysis of the phase diagram is deferred to another
work.

###  IV.6 Discussions

We are not aware of conclusive evidence for the existence of the
unconventional magnets. Taking into account the great discoveries of the
unconventional superconductivity and pairing superfluidity in high Tc cuprates
and 3He, respectively, we are optimistic that unconventional magnetic phases
also exist in Nature. We propose to systematically search for these new phases
in 3He, ultracold atomic systems, semiconductors, heavy fermion materials and
ruthenates, both in experiments and in numerical simulations.

Unconventional magnetic orders are natural generalizations of itinerant
ferromagnetism, whose driving force is still the exchange interaction. But it
needs to be in the non-local version, i.e., a non-$s$-wave channel.
Nevertheless, interactions in the high angular momentum channels are typically
weak. In Ref. Lee and Wu (2009), a heuristic argument is provided to employ
the orbital hybridized band structure to promote the Landau interaction to
high partial-wave channels. Consider a $d_{xz}/d_{yz}$ hybridized orbital
band. Around the Fermi surface, the Bloch wavefunction takes the orbital
configuration as

$\displaystyle|\Psi_{\alpha}(\mathbf{k})\rangle$ $\displaystyle=$
$\displaystyle
e^{i\mathbf{k}\mathbf{r}}\left(\cos\phi_{k}|d_{xz}\rangle+\sin\phi_{k}|d_{yz}\rangle\right)\otimes\chi_{\alpha},$
(132)

where $\chi_{\alpha}$ is the spin eigenstate. The Landau interaction at the
Hartree-Fock level is

$\displaystyle f_{\uparrow\uparrow}(\mathbf{k}_{1}\mathbf{k}_{2})$
$\displaystyle=$ $\displaystyle V(\mathbf{q}=0)-\frac{1}{2}(1+\cos
2\theta_{\mathbf{k}_{1}\mathbf{k}_{2}})V(\mathbf{k}_{1}-\mathbf{k}_{2}),$
$\displaystyle f_{\uparrow\downarrow}(\mathbf{k}_{1}\mathbf{k}_{2})$
$\displaystyle=$ $\displaystyle V(\mathbf{q}=0).$ (133)

The appearance of the $d$-wave form factor $\cos
2\theta_{\mathbf{k}_{1}\mathbf{k}_{2}}$ is due to the orbital hybridization,
i.e., even though two electrons possess the same spin, they can still be
distinguished by their orbital components. Hence, although
$V(\mathbf{k}_{1}-\mathbf{k}_{2})$ could be dominated by the $s$-wave
component, the angular form factor shifts a significant part of the spectral
weight into the $d$-wave channel. Based on this formalism, a possible
explanation of the nematic transition observed in Sr3Ru2O7 was provided.

Below we summarize several possible directions for searching unconventional
magnetism. Ferromagnetic fluctuations in the normal state of 3He are strong.
The values of $F_{1a}$ of 3He are measured as negative via the normal-state
spin diffusion constant, spin-wave spectrum, and the temperature dependence of
the specific heat Leggett (1970); Corruccini  _et al._ (1971); Osheroff
(1977); Greywall (1983). It varies from around $-0.5$ to $-1.2$ with
increasing pressures to the melting point, reasonably close to the instability
point $F^{a}_{1}=-3$. We conjecture that 3He could support unconventional
magnetism under certain conditions or exhibit strong fluctuations of these
orders.

An important direction to search for unconventional magnetism is the so-called
“hidden-order” systems. Hidden orders typically mean that thermodynamic
quantity measurements exhibit a transition to a low temperature ordered state.
However, the nature of the orders remains unknown since they do not exhibit
themselves in typical detections. Unconventional magnetic orders neither break
translation symmetry nor exhibit magnetic orderings in real space. They are
multipolar orderings in momentum space, hence, they are difficult to detect
via typical experimental methods. Hence, they are natural candidates for
hidden orders. In fact, multipolar orderings in real space are also popular
candidates for hidden orders in literature .

For example, the well-known system of heavy fermion compound URu2Si2 exhibits
a mysterious phase transition at 17K by showing a large anomaly in specific
heat. It also exhibits a jump in the non-linear magnetic susceptibility at the
transition. However, even with efforts after a few decades, the nature of this
transition remains elusive Mydosh  _et al._ (2020); Wolowiec  _et al._ (2021).
Varma proposed an order, which is essentially the $p$-wave $\alpha$-phase in
our language Varma (2005); Varma and Zhu (2006). Calculations for
thermodynamic quantities fit in experiment measurements reasonably well.
Another hidden order compound Cd2Re2O7 exhibits heat capacity anomaly and a
kink of DC resistivity around 200K. Recently, it has been discovered that the
hidden order phase exhibits inversion symmetry breaking via the optical 2nd
harmonic generation measurements Harter  _et al._ (2017, 2018). Since this is
a heavy element compound, Pomeranchuk instabilities of spin-orbit coupled
Fermi liquid theory may be a promising candidate Fu (2015); Norman (2020).

An obstacle to identifying unconventional magnetism is the lack of definitive
experimental signatures and detection methods. We know that antiferromagnetism
is very common among transition metal oxides, more common than itinerant
ferromagnetism. However, the experimental identification of the
antiferromagnetic ordering is only possible after the detection method of
neutron scattering spectroscopy became available.

Maybe unconventional magnetism already exists somewhere, but we need to think
about how to detect them. In addition to the inelastic neutron scattering
resonances (Sect. IV.4.3), we outline the following possible methods.

The $\beta$ phases exhibit effective spin-orbit coupling, hence, standard
methods to detect spin-orbit coupling still apply. The distinctive feature is
that the spin-orbit coupling effects should turn on and off at a phase
transition.

Transport measurements can be used to detect the dynamic generation of spin-
orbit coupling. For example, the existence of the anomalous Hall effect (AHE)
relies on spin-orbit coupling. Therefore, detecting the AHE signal turning on
at a phase transition would be an evidence of the onset of the entanglement of
spin and momentum. As for the $d$-wave $\alpha$-phase, i.e., spin-$\uparrow$
and spin-$\downarrow$ Fermi surfaces exhibit opposite quadrupolar distortions.
Taking the principle axes of the quadrupolar distortion as $x$ and $y$-axis,
it is straightforward to show that the spin and charge currents satisfy

$\displaystyle\left(\begin{array}[]{c}j_{x}^{sp}\\\
j_{y}^{sp}\end{array}\right)\propto\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\left(\begin{array}[]{c}j_{x}^{c}\\\
j_{y}^{c}\end{array}\right).$ (140)

A verification of this transport relation would be a signature of the $d$-wave
$\alpha$-phase Wu  _et al._ (2007).

Methods that can detect Fermi surface splitting are useful. The angular
resolved photon emission spectroscopy (ARPES) can be used to detect the band
splitting. In fact, such an experiment has been performed in the system with
relativistic spin-orbit coupling. In the unconventional magnetic phases, ARPES
in principle can measure temperature-dependent Fermi surface splittings. Fermi
surface splitting also shows up in quantum oscillation experiments (e.g.
Shubunikov-de Haas (SdH) oscillations) as beat patterns. Hence, a temperature-
dependent beat pattern in this kind of experiments would be a signature of the
developing of unconventional magnetism.

##  V Conclusions

We have reviewed a few applications of the symmetry principle in condensed
matter and cold atom systems.

First, we reviewed the concept of “space-time” group, which provides a
symmetry framework in studying transport and topological properties in a
variety of dynamic systems beyond the Floquet framework, such as laser-driven
solid state lattices, dynamic photonic crystals, and optical lattices. Various
fundamental concepts are generalized, including space-time unit cell,
momentum-energy Brillouin zone, Bloch-Floquet theory. Novel non-symmorphic
space-time transformations are identified including time-screw rotation, time-
glide reflection, and time-shift rotary reflection. 13 space-time groups are
classified in 1+1D with 5 of them non-symmorphic, and 275 space-time group are
classified in 2+1D. We expect that space-time group will play an important
role for studying dynamic systems, in a similar way to space group for static
crystals.

Second, we reviewed the progress of studying large-spin ultracold fermions
from the perspective of high symmetries. Due to enhanced quantum spin
fluctuations from the large number of fermion components, such systems
naturally lie in the large-$N$ region instead of the large-$S$ region which is
typically studied in solids. A generic Sp(4), or, isomorphically, SO(5)
symmetry is proved in spin-$\frac{3}{2}$ systems, which plays a similar role
of SU(2) in spin-$\frac{1}{2}$ systems. This symmetry can be upgraded to SO(7)
under certain conditions which extends Yang’s $\eta$-pairing to $\chi$-pairing
as its high rank Lie algebra counterpart. The 7D vector and 21D adjoint
representations of SO(7) unify a variety of competing orders in both particle-
particle and particle-hole channels. Large-spin systems can exhibit multi-
particle clustering orderings or correlations both in the superfluid state
with attractions and in the super-exchange physics with repulsions, which is
similar to 3-quark baryon (color singlet) formation in high energy physics.
The competitions among quarteting superfluidity/density-wave and pairing
superfluidity/density-wave are investigated. The SU(4) singlet plaquette
states in a 3D cubic lattice can be described by a quantum plaquette model,
whose effective description is mapped to a high order gauge theory. We
anticipate that research along this direction can bridge cold atom physics,
condensed matter matter, and high energy physics together. Along with the
experimental progress, even more exotic strong coupling physics that is not
easily accessible in usual solid state systems could be investigated.

At last, we reviewed the unconventional magnetism as a mechanism of “spin from
isospin” to generate spin-orbit coupling in non-relativistic Fermi liquids.
They are also novel states of itinerant electrons generalzing ferromagnetism
to unconventional symmetries based on the Fermi surface instabilities of the
Pomeranchuk type. These states include the isotropic $\beta$-phase and the
anisotropic $\alpha$-phase, which are the particle-hole channel analogy to the
superfluid 3He-B and A phases, respectively. Different from the relativistic
spin-orbit coupling, these dynamically generated spin-orbit couplings possess
collective excitations of Goldstone modes, whose dynamics couples to spin
moment and induces resonances in the inelastic neutron scattering
spectroscopy. Possible realizations of “unconventional magnetism” in hidden
order systems and experimental detections are discussed.

##  VI Acknowledgments

I am grateful to my Ph. D. advisor, S. C. Zhang (deceased), who convinced me
of the beauty and the power of symmetry. The last two topics reviewed here
were started in my Ph. D. period and were continued with various new
developments in my career. I thank Jiangping Hu, E. H. Fradkin, K. Sun, D.
Arovas, W. C. Lee, H. H. Hung, Shu Chen, Yupeng Wang, C. K. Xu, Y. Li, D.
Wang, S. L. Xu, Z. Q. Gao, M. Pan, C. H. Ke, Z. X. Lin for collaborations on
related topics, and J. E. Hirsch, A. L. Fetter, T. L. Ho, S. Das Sarma, S.
Kivelson, L. J. Sham for their warm encouragements and appreciations. I also
thank Ji Wang for proofreading and polishing. This work is supported by NSFC
under the Grants No. 12174317 and No. 11729402.

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