# Tiltan and Superclub

Shimon Garti  Einstein Institute of Mathematics, The Hebrew University of
Jerusalem, Jerusalem 91904, Israel<EMAIL_ADDRESS>and  Saharon
Shelah  Institute of Mathematics The Hebrew University of Jerusalem Jerusalem
91904, Israel and Department of Mathematics Rutgers University New Brunswick,
NJ 08854, USA<EMAIL_ADDRESS>http://www.math.rutgers.edu/~shelah

###### Abstract.

We show that one can force superclub with an arbitrarily large value of ${\rm
cov}(\mathscr{M})$. We prove that the club principle is consistent with an
arbitrarily large value of ${\rm add}(\mathscr{M})$. We also prove that if
$\kappa$ is regular then superclub at $\kappa^{+}$ implies
$Q(\kappa^{+},\kappa^{+},\kappa^{+})$.
Nous prouvons que superclub est consistant avec un valeur arbitrairement
élevée de ${\rm cov}(\mathscr{M})$. Nous prouvons que trèfle est consistant
avec un valeur arbitrairement élevée de ${\rm add}(\mathscr{M})$. Nous
prouvons aussi que superclub en $\kappa^{+}$ implique
$Q(\kappa^{+},\kappa^{+},\kappa^{+})$ si $\kappa$ est cardinal regulier.

###### Key words and phrases:

Superclub, club (tiltan), invariants of measure and category, infinite graphs,
square brackets

###### 2010 Mathematics Subject Classification:

05C63, 03E02, 03E17

Research supported by ISF grant no. 1838/19. This is publication 1179 in the
second author’s list

##  0\. Introduction

The diamond principle $\Diamond_{\kappa}$ is a prediction principle discovered
by Jensen in [Jen72]. Suppose that $\kappa$ is regular and uncountable. A
$\Diamond_{\kappa}$-sequence is a sequence of sets
$(A_{\alpha}:\alpha\in\kappa)$ such that $A_{\alpha}\subseteq\alpha$ whenever
$\alpha\in\kappa$ and for every $A\subseteq\kappa$ the set
$S_{A}=\\{\alpha\in\kappa:A\cap\alpha=A_{\alpha}\\}$ is a stationary subset of
$\kappa$. We say that $\Diamond_{\kappa}$ holds iff there exists a
$\Diamond_{\kappa}$-sequence. Notice that $\Diamond_{\kappa^{+}}$ implies
$2^{\kappa}=\kappa^{+}$.

The club principle $\clubsuit_{\kappa}$ is a weaker prediction principle,
discovered by Ostaszewski in [Ost76]. Denote the set of limit ordinals of
$\kappa$ by $\lim(\kappa)$. We say that $(T_{\delta}:\delta\in\lim(\kappa))$
is a $\clubsuit_{\kappa}$-sequence iff every $T_{\delta}$ is an unbounded
subset of $\delta$ and for every $A\in[\kappa]^{\kappa}$ the set
$T_{A}=\\{\delta\in\lim(\kappa):T_{\delta}\subseteq A\cap\delta\\}$ is a
stationary subset of $\kappa$.

The difference between $\Diamond_{\kappa}$ and $\clubsuit_{\kappa}$ is two-
fold. Firstly, $\Diamond_{\kappa}$ is based on equality while
$\clubsuit_{\kappa}$ is based on inclusion. Secondly, $\Diamond_{\kappa}$
predicts every element of $\mathcal{P}(\kappa)$ while $\clubsuit_{\kappa}$
predicts only elements of $[\kappa]^{\kappa}$. The latter property is
responsible for the fact that $\clubsuit_{\kappa^{+}}$ is consistent with
$2^{\kappa}>\kappa^{+}$. Since $\Diamond_{\kappa^{+}}$ implies
$2^{\kappa}=\kappa^{+}$ one can see that $\clubsuit_{\kappa^{+}}$ is strictly
weaker than $\Diamond_{\kappa^{+}}$.

Motivated by a stubborn open problem of Juhasz about the connection between
$\clubsuit_{\aleph_{1}}$ and the existence of Suslin trees, Primavesi defined
in [Pri11] an intermediate principle, dubbed as _superclub_. We phrase the
definition in the generalized form.

######  Definition 0.1.

Superclub.
Assume that $\kappa={\rm cf}(\kappa)>\aleph_{0}$.

  1. $(\aleph)$

A superclub sequence at $\kappa$ is a sequence of sets
$(S_{\alpha}:\alpha\in\lim(\kappa))$ such that $S_{\alpha}$ is an unbounded
subset of $\alpha$ for each $\alpha$ and for every $A\in[\kappa]^{\kappa}$
there exists $B\in[A]^{\kappa}$ so that
$S_{B}=\\{\alpha\in\lim(\kappa):B\cap\alpha=S_{\alpha}\\}$ is a stationary
subset of $\kappa$.

  2. $(\beth)$

Superclub holds at $\kappa$ iff there exists a superclub sequence for
$\kappa$.

Comparing superclub to diamond and club it is clear from the definition that
diamond at $\kappa$ implies superclub at $\kappa$ which implies, in turn, the
club principle at $\kappa$. Both implications are irreversible. It has been
proved in [Gar18] that superclub at $\kappa^{+}$ implies Galvin’s property at
$\kappa^{+}$ while $\clubsuit_{\kappa^{+}}$ is consistent with the failure of
Galvin’s property at $\kappa^{+}$. It has been proved in [Che17] that
superclub at $\aleph_{1}$ is consistent with $2^{\aleph_{0}}>\aleph_{1}$ while
$\Diamond_{\aleph_{1}}$ implies $2^{\aleph_{0}}=\aleph_{1}$.

The forcing of [Che17] is based on the existence of a strongly inaccessible
cardinal in the ground model. A natural question is whether superclub with a
large continuum has some consistency strength. We shall give a negative
answer. Specifically, in the model of [FSS97] the continuum hypothesis fails
and superclub holds. An immediate consequence is the consistency of superclub
with a large value of some cardinal characteristics like the dominating number
$\mathfrak{d}$.

A more interesting challenge is the bounding number $\mathfrak{b}$ and the
category invariant ${\rm add}(\mathscr{M})$. We do not know whether superclub
is consistent with a large value of these characteristics, even if the desired
value is just $\omega_{2}$. However, we shall force club with an arbitrarily
large value of $\mathfrak{b}$ and ${\rm cov}(\mathscr{M})$ simultaneously,
thus we will have a large value of ${\rm add}(\mathscr{M})$ as well.

From the definition it seems that superclub has more affinities with the club
principle. The prediction of superclub applies only to $[\kappa]^{\kappa}$ and
based on inclusion since if $B\cap\alpha=S_{\alpha}$ then $S_{\alpha}\subseteq
A\cap\alpha$ and possibly $S_{\alpha}\neq A\cap\alpha$. Hence in the above
mentioned features which distinguish club from diamond, superclub behaves like
club and unlike diamond. But there is another feature in which superclub is
similar to diamond.

If $A\subseteq\kappa$ and $S_{A}$ is the stationary set of guesses given by
$\Diamond_{\kappa}$ then the corresponding elements of the diamond sequence
are coherent in the following sense. If $\gamma,\delta\in S_{A}$ and
$\gamma<\delta$ then $A_{\delta}$ end-extends $A_{\gamma}$. This is a
consequence of the equality which forms the prediction of diamond sequences.
If one assumes only club at $\kappa$ then this coherence fades away. Actually,
one can choose a club sequence and then shrink every element in the sequence
by taking subsets of order-type $\omega$, in which case $A_{\delta}$ never
extends $A_{\gamma}$.

From this point of view, but only from this one, superclub is similar to
diamond. Given $A\in[\kappa]^{\kappa}$ and choosing $B\in[A]^{\kappa}$, the
superclub sequence acts on $B$ as a diamond sequence with equality as the
predicate for prediction. Consequently, the elements of the superclub sequence
are coherent, despite the fact that they are only included in the original set
$A$. In other words, a superclub sequence is a coherent club sequence. This
idea stands behind the proof of Galvin’s property from superclub, and it will
be exploited in the last section.

Our notation is (hopefully) standard. We shall use the Jerusalem forcing
notation, so $p\leq q$ means that $p$ is weaker than $q$. Occasionally we
shall use the word _tiltan_ instead of _club_ when refering to the club
principle. The paper contains three additional sections. In the first part we
force superclub with large continuum without an inaccessible cardinal.
Actually, ${\rm cov}(\mathscr{M})$ assumes the value of the continuum in the
generic extension, hence $\mathfrak{d}=2^{\omega}$ as well. In the second part
we show that one can force club with ${\rm add}(\mathscr{M})=\mathfrak{c}$
where $\mathfrak{c}$ is arbitrarily large. In the last section we deal with a
graph-theoretic statement which follows from the continuum hypothesis and here
we show that it follows merely from superclub.

We thank the anonymous referee of the paper for a careful reading of the
article and for the helpful comments, all of them are integrated within the
manuscript. We also thank Thilo Weinert for a helpful conversation concerning
Cichoń’s diagram and the bounding number. Finally, we thank Jörg Brendle and
Lajos Soukup for several important comments and useful information.

##  1\. A takeaway theorem

In this section we prove that the method of [FSS97] for enlarging the
continuum while preserving club sequences from the ground model works equally
well with respect to superclub sequences. The main conclusion is that one can
force superclub with an arbitrarily large value of the continuum without the
assumption that there is an inaccessible cardinal in the ground model. Another
conclusion is that one can force superclub with a large value of ${\rm
cov}(\mathscr{M})$, a result which will be useful in the next section.

Let us describe the basic forcing notion of [FSS97] in the spirit of Prikry
forcing, namely we define the forcing order which is quite incomplete and we
also define a pure order which is $\omega_{1}$-complete.

######  Definition 1.1.

The Fuchino-Shelah-Soukup forcing.
Let $(\mathbb{P},\leq,\leq^{*})$ be the following forcing notion.

  1. $(\aleph)$

A condition $p\in\mathbb{P}$ is a partial function from $\omega_{2}$ into
$\\{0,1\\}$ such that $|{\rm dom}(p)|\leq\aleph_{0}$ and ${\rm
dom}(p)\cap[\delta,\delta+\omega)$ is finite whenever $\delta$ is a limit
ordinal of $\omega_{2}$.

  2. $(\beth)$

If $p,q\in\mathbb{P}$ then
$u_{pq}=\\{\delta\in\lim(\omega_{2}):\varnothing\neq{\rm
dom}(p)\cap[\delta,\delta+\omega)={\rm dom}(q)\cap[\delta,\delta+\omega)\\}$.

  3. $(\gimel)$

If $p,q\in\mathbb{P}$ then $p\leq q$ iff $p\subseteq q$ and $u_{pq}$ is
finite. We define $p\leq^{*}q$ iff $p\subseteq q$ and $u_{pq}$ is empty.

The following statement is proved in [Pri11, Proposition 5.1.4]. If
$\textstyle f$ $\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ is a
$\mathbb{P}$-name, $p\in\mathbb{P}$ and $p$ forces that
$\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}:\omega_{1}^{V}\rightarrow\omega_{1}^{V}$ is a function then
one can find a set $A_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\in[\omega_{1}]^{\omega_{1}}\cap V$ and a function
$g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}:A_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\rightarrow\omega_{1}$ such that
$g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\in V$ and for every ordinal $\eta\in\omega_{1}^{V}$ there
is a condition $q_{\eta}\in\mathbb{P}$ such that $p\leq q_{\eta}$ and
$q_{\eta}\Vdash g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\upharpoonright(A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\cap\eta)=\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\upharpoonright(A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\cap\eta)$. Similar assertions appear in [FSS97], see for
example Lemma 3.9 there.

It follows from this statement that $\aleph_{1}$ is preserved, and moreover
stationary subsets of $\aleph_{1}$ remain stationary in the generic extension
as follows from [FSS97, Theorem 2.6]. To see that $\aleph_{1}$ is preserved
consider $\mathchoice{\oalign{$\displaystyle h$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle h$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle h$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle h$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}:\omega_{1}^{V}\rightarrow\omega$ and a condition $p$ which
forces this fact. Let $g$ be $g_{p,\mathchoice{\oalign{$\displaystyle
h$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle h$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle h$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle h$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}$, so ${\rm rang}(g)\subseteq\omega$. Since
$A_{p,\mathchoice{\oalign{$\displaystyle h$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle h$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle h$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle h$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}={\rm dom}(g)$ is uncountable, there are $\beta,\gamma\in
A_{p,\mathchoice{\oalign{$\displaystyle h$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle h$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle h$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle h$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}$ for which $g(\beta)=g(\gamma)$. Now if
$\delta\in\omega_{1}^{V}$ is sufficiently large then $q_{\delta}\geq p$ and
$q_{\delta}\Vdash\mathchoice{\oalign{$\displaystyle h$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle h$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle h$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle h$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}(\beta)=g(\beta)=g(\gamma)=\mathchoice{\oalign{$\displaystyle
h$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle h$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle h$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle h$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}(\gamma)$ so  $\textstyle h$
$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ cannot be one to one and
hence $\omega_{1}^{V}$ is not a countable ordinal in $V[G]$.

In the above definition the domain of any condition $p$ is contained in
$\omega_{2}$, but one can replace $\omega_{2}$ by an arbitrarily large
cardinal $\kappa$. If $\kappa^{\aleph_{0}}=\kappa$ in the ground model then
$2^{\omega}=\kappa$ in the generic extension.

######  Theorem 1.2.

Let $V$ be a model of GCH and let $(S_{\alpha}:\alpha\in\lim(\omega_{1}))$ be
a superclub sequence in $V$. Let $\kappa\geq\omega_{2}$ be such that
$\kappa^{\aleph_{0}}=\kappa$, let $\mathbb{P}$ be the Fuchino-Shelah-Soukup
forcing based on $\kappa$, and let $G\subseteq\mathbb{P}$ be generic over $V$.
Then cardinals and cofinalities are preserved in $V[G]$, the continuum becomes
$\kappa$ and $(S_{\alpha}:\alpha\in\lim(\omega_{1}))$ remains a superclub
sequence in $V[G]$. Hence superclub is consistent with an arbitrarily large
value of ${\rm cov}(\mathscr{M})$.

_Proof_.
By the comment before the statement of the theorem we know that $\aleph_{1}$
is preserved. Cardinals above $\aleph_{1}$ are preserved since $\mathbb{P}$ is
$\aleph_{2}$-cc (here we use the fact that $V$ models GCH). It is easy to
infer from density arguments that $2^{\omega}=\kappa$ in $V[G]$, as the
generic set $G$ adds a function from $\kappa$ into $\\{0,1\\}$ which can be
sliced into $\kappa$ distinct functions from $\omega$ into $\\{0,1\\}$.

Suppose that $A\in[\omega_{1}]^{\omega_{1}}\cap V[G]$ and let  $\textstyle f$
$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ be a name for the increasing
enumeration of the elements of $A$. Fix a condition $p$ which forces this
fact. Let $A_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}$ and $g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}$ be as described ahead of the proof, so
$A_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\in V$ and for every $\delta\in\omega_{1}^{V}$ one can find
$q_{\delta}\geq p$ such that $q_{\delta}\Vdash{\rm
rang}(g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\upharpoonright A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}})\subseteq A$. Applying superclub in the ground model to
$g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}^{\prime\prime}A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}$, let $B$ be an uncountable subset of
$g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}^{\prime\prime}A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}$ on which the superclub sequence
$(S_{\alpha}:\alpha\in\lim(\omega_{1}))$ acts like diamond.

Let $S_{B}=\\{\alpha\in\omega_{1}:B\cap\alpha=S_{\alpha}\\}$. If $\alpha\in
S_{B}$ then choose $\delta\in(\alpha,\omega_{1})$ such that
$S_{\alpha}\subseteq{\rm rang}(g_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\upharpoonright(A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\cap\delta))$. Let $q_{\delta}\geq p$ be so that
$q_{\delta}\Vdash g_{p,\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\upharpoonright(A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\cap\delta)=\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\upharpoonright(A_{p,\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}\cap\delta)$. It follows that $q_{\delta}\Vdash
B\cap\alpha=S_{\alpha}$. Since this is true for every $\alpha\in S_{B}$ and
the stationarity of $S_{B}$ is preserved, one can see that superclub holds in
the generic extension. It is easy to verify that ${\rm cov}(\mathscr{M})$
equals $\mathfrak{c}$ in the generic extension, see [Bre06] for details, so we
are done.

$\qed_{\ref{thmaway}}$

Remark that the main result of this section already shows that in order to
force superclub with a large value of the continuum there is no need of an
inaccessible cardinal in the ground model. In the next section we will show
that the above result can be strengthened if one replaces superclub by tiltan.
The idea is that the forcing notion of [FSS97] can be composed with Hechler
forcing, thus making both ${\rm cov}(\mathscr{M})$ and $\mathfrak{b}$ large.
We do not know how to preserve superclub sequences from the ground model in
this construction, but if one begins with a diamond sequence in the ground
model then it remains a club sequence in the generic extension.

##  2\. The additivity of the meager ideal

In this section we prove the consistency of tiltan (the club principle) with
an arbitrarily large value of ${\rm add}(\mathscr{M})$. This confirms a
conjecture of Brendle, see [Bre06, Conjecture 9.4]. The context of this result
is nicely displayed by Cichoń’s diagram:

---
$\textstyle{{\rm
cov}(\mathscr{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
non}(\mathscr{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
cof}(\mathscr{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
cof}(\mathscr{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
add}(\mathscr{N})}$$\textstyle{{\rm
add}(\mathscr{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
cov}(\mathscr{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
non}(\mathscr{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

A result of Truss from [Tru83] says that $\clubsuit_{\aleph_{1}}$ implies
${\rm add}(\mathscr{N})=\aleph_{1}$, so the smallest invariant in the above
diagram becomes $\aleph_{1}$ under the club principle. Nevertheless, we shall
see that $\clubsuit_{\aleph_{1}}$ is consistent with an arbitrarily large
value of ${\rm add}(\mathscr{M})$. The idea is to increase both ${\rm
cov}(\mathscr{M})$ and $\mathfrak{b}$ while keeping the club principle. By a
result of Miller from [Mil81] we know that ${\rm
add}(\mathscr{M})=\min(\mathfrak{b},{\rm cov}(\mathscr{M}))$, so our theorem
will follow.

A natural way to increase ${\rm cov}(\mathscr{M})$ is by using [FSS97], as
described in the previous section. A natural way to increase $\mathfrak{b}$ is
Hechler forcing from [Hec74]. We need, therefore, to amalgamate these two
forcing notions. Moreover, we would like to do it while maintaining tiltan.
For this end, we shall use a strong version of $ccc$, called _sweetness_. The
concept of sweet forcing notions comes from [She84], and a good background is
contained in [RaS04]. Let us recall the formal definition.

######  Definition 2.1.

Sweet forcing notions.
A forcing notion $\mathbb{P}$ is sweet iff there are a dense subset
$\mathcal{D}$ of $\mathbb{P}$ and a sequence $\bar{E}=(E_{n}:n\in\omega)$ of
equivalence relations on $\mathcal{D}$ such that the following requirements
are met:

  1. $(a)$

Each $E_{n}$ is $\leq_{\mathbb{P}}$-directed and $\mathcal{D}/E_{n}$ is
countable.

  2. $(b)$

$E_{n+1}\subseteq E_{n}$ for every $n\in\omega$.

  3. $(c)$

If $\\{p_{i}:i\in\omega\\}\subseteq\mathcal{D}$ and $r\in\mathcal{D}$
satisfies $p_{i}E_{i}r$ for every $i\in\omega$ then for every $n\in\omega$
there is a condition $q_{n}\geq r$ such that $q_{n}E_{n}r$ and $\forall i\geq
n,p_{i}\leq q_{n}$.

  4. $(d)$

If $p,q\in\mathcal{D}$ and $n\in\omega$ then there is some $m\in\omega$ so
that for every $r\in[p]_{E_{m}}$ one can find $t\in[q]_{E_{n}}$ such that
$r\leq t$.

There is another concept of sweetness, defined by Stern in [Ste85] and based
on topological considerations. For the formal definition of topological
sweetness and the concept of an iterably sweet forcing notion we refer to
[RaS04, Section 4]. Here we just mention the fact that if $\mathbb{P}$ is
sweet and every pair of compatible conditions has a least upper bound then
$\mathbb{P}$ is iterably sweet. The forcing notion of [FSS97] mentioned in the
previous section is sweet and also iterably sweet. It has been proved in
[She84, Section 7] that if $\mathbb{P}$ is sweet and  $\textstyle\mathbb{D}$
$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ is a $\mathbb{P}$-name of
Hechler forcing then
$\mathbb{P}\ast\mathchoice{\oalign{$\displaystyle\mathbb{D}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{D}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{D}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{D}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}$ is sweet as well.

In what follows we describe a class of forcing notions which generalize the
forcing of [FSS97]. Define $\mathcal{K}_{0}$ as the class of objects $K$ which
are essentially an iteration of $ccc$ forcing notions. So $\ell g(K)$ is an
ordinal (the length of the iteration) and $\mathbb{P}_{\alpha}$ is a forcing
notion for every $\alpha\leq\ell g(K)$. If $\alpha\leq\beta\leq\ell g(K)$ then
we require $\mathbb{P}_{\alpha}\lessdot\mathbb{P}_{\beta}$, and for every
$\beta<\ell g(K)$ we have a $\mathbb{P}_{\beta}$-name
$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\beta}$ of a $ccc$ forcing notion.

Suppose that $\alpha\leq\ell g(K)$. A condition $p\in\mathbb{P}_{\alpha}$ is a
function so that ${\rm dom}(p)\subseteq\alpha$ and $|{\rm
dom}(p)|\leq\aleph_{0}$. If $\beta\in{\rm dom}(p)$ then $p(\beta)$ is a
$\mathbb{P}_{\beta}$-name of an element of
$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\beta}$ and if $\beta<\alpha$ then
$p\upharpoonright\beta\in\mathbb{P}_{\beta}$. If $p,q\in\mathbb{P}_{\alpha}$
then $p\leq_{\mathbb{P}_{\alpha}}q$ iff ${\rm dom}(p)\subseteq{\rm dom}(q)$
and for every $\beta\in{\rm dom}(p)$ we have
$p\upharpoonright\beta\Vdash_{\mathbb{P}_{\beta}}p(\beta)\leq_{\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}}q(\beta)$, and the set ${\rm diff}(p,q)=\\{\beta\in{\rm
dom}(p):p(\beta)\neq q(\beta)\\}$ is finite.

Apart from the usual ordering $\leq_{\mathbb{P}_{\alpha}}$ we define the
orderings $\leq^{\rm pr}_{\alpha}$ and $\leq^{\rm ap}_{\alpha}$ where pr
stands for pure and ap stands for apure. Formally, $\leq^{\rm
pr}_{\alpha}=\\{(p,q)\in\mathbb{P}_{\alpha}\times\mathbb{P}_{\alpha}:p\leq_{\mathbb{P}_{\alpha}}q\wedge
q\upharpoonright{\rm dom}(p)=p\\}$ and $\leq^{\rm
ap}_{\alpha}=\\{(p,q)\in\mathbb{P}_{\alpha}\times\mathbb{P}_{\alpha}:p\leq_{\mathbb{P}_{\alpha}}q\wedge{\rm
dom}(p)={\rm dom}(q)\\}$. Notice that if $p\leq_{\mathbb{P}_{\alpha}}r$ then
one can separate the extension by finding $q\in\mathbb{P}_{\alpha}$ so that
$p\leq^{\rm pr}_{\alpha}q\leq^{\rm ap}_{\alpha}r$. If $K\in\mathcal{K}_{0}$
and $\alpha=\ell g(K)$ then
$\leq_{K}=\leq_{\mathbb{P}_{\alpha}},\mathbb{P}_{K}$ denotes the whole
iteration and so forth.

Assuming CH in the ground model, if $K\in\mathcal{K}_{0}$ then
$\mathbb{P}_{K}$ is $\aleph_{2}$-cc. The pure order is $\aleph_{1}$-complete,
and moreover if $(p_{n}:n\in\omega)$ is $\leq_{K}^{\rm pr}$-increasing then
$\bigcup_{n\in\omega}p_{n}$ is an upper bound, which will be called the
_canonical_ upper bound. One can verify that $\mathbb{P}_{K}$ is proper.
Assume that $(p_{i}:i\in\omega_{1})$ is $\leq^{\rm pr}_{K}$-increasing and
continuous, that is $p_{j}$ is the canonical upper bound of $(p_{i}:i\in j)$
whenever $j\in\omega_{1}$ is a limit ordinal. Assume further that
$p_{i+1}\leq^{\rm ap}_{K}q_{i+1}$ for every $i\in\omega_{1}$. Then one can
find $i<j<\omega_{1}$ such that $q_{i+1}\parallel q_{j+1}$. It can be verified
that the forcing notion of [FSS97] used in the previous section belongs to
$\mathcal{K}_{0}$. Let $\mathcal{K}$ be the subclass of $\mathcal{K}_{0}$
which consists of iterations $K$ which are Suslin $ccc$ (this means that the
forcing and its order are $\Sigma^{1}_{1}$-definable) and iterably sweet. We
shall use this class in order to prove the following:

######  Theorem 2.2.

Assume that $\Diamond_{\aleph_{1}}$ holds in $V,K\in\mathcal{K}$ and
$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{K,\beta}$ is a name of Hechler forcing for every
$\beta\leq\ell g(K)$. Then $\clubsuit_{\aleph_{1}}$ holds in the generic
extension by $\mathbb{P}_{K}$. Concomitantly, $\mathfrak{b}=\ell g(K)$ in the
generic extension by $\mathbb{P}_{K}$, provided that $\ell g(K)$ is a regular
and uncountable cardinal.

_Proof_.
The fact that $\mathfrak{b}=\ell g(K)$ in the generic extension by
$\mathbb{P}_{K}$ follows from the properties of Hechler forcing. We must show,
therefore, that $\clubsuit_{\aleph_{1}}$ holds in this generic extension. Let
$(A_{\alpha}:\alpha\in\omega_{1})$ be a diamond sequence in $V$. Since $\ell
g(K)$ is typically large, we get the negation of CH in the generic extension
and hence diamond fails. However, we will show that
$(A_{\alpha}:\alpha\in\omega_{1})$ exemplifies $\clubsuit_{\aleph_{1}}$ in the
generic extension. Suppose that  $\textstyle A$
$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ is a $\mathbb{P}_{K}$-name of
an element of $[\omega_{1}]^{\aleph_{1}}$. Fix a generic set
$G\subseteq\mathbb{P}_{K}$ and a condition $p$ such that
$p\Vdash\mathchoice{\oalign{$\displaystyle A$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle A$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle A$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle A$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\in[\omega_{1}]^{\aleph_{1}}$. By induction on
$i\in\omega_{1}$ we choose a triple $(p_{i},q_{i},\alpha_{i})$ such that the
following requirements are met:

  1. $(a)$

$p_{0}=p$.

  2. $(b)$

$\alpha_{i}\in\omega_{1}$.

  3. $(c)$

$(p_{j}:j\leq i)$ is $\leq_{K}^{\rm pr}$-increasing and continuous.

  4. $(d)$

$p_{i}\leq_{K}q_{i}$.

  5. $(e)$

$q_{i}$ forces that the $i$th member of  $\textstyle A$
$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ is $\check{\alpha}_{i}$.

  6. $(f)$

$p_{i+1}\leq_{K}^{\rm ap}q_{i}$.

Let $\chi$ be a sufficiently large regular cardinal. For every
$\varepsilon\in\omega_{1}$ choose $N_{\varepsilon}\prec\mathcal{H}(\chi)$ such
that each $N_{\varepsilon}$ is countable,
$(N_{\varepsilon}:\varepsilon\in\omega_{1})$ is increasing and continuous,
$(N_{\zeta}:\zeta\leq\varepsilon)\in N_{\varepsilon+1},K\in N_{0}$ and
$((p_{i},q_{i},\alpha_{i}):i\in\omega_{1})\in N_{0}$. Fix an ordinal
$\delta\in\omega_{1}$ so that $N_{\delta}\cap\omega_{1}=\delta$.

Let $w={\rm diff}(p_{\delta+1},q_{\delta})=\\{\alpha\in{\rm
dom}(q_{\delta}):p_{\delta+1}(\alpha)\neq q_{\delta}(\alpha)\\}$. Choose a
sequence $(\varepsilon_{n}:n\in\omega)$ such that
$m<n\Rightarrow\varepsilon_{m}<\varepsilon_{n},\delta=\bigcup_{n\in\omega}\varepsilon_{n}$
and the following property holds. For every first order formula
$\varphi(x,\bar{y})$ and every $\bar{a}\in{}^{\ell g(\bar{y})}N_{\delta}$ and
every large enough
$n\in\omega,\mathcal{H}(\chi)\models\varphi[q_{\delta},\bar{a}]\equiv\varphi[q_{\varepsilon_{n}},\bar{a}]$.
For each $n\in\omega$ let $\delta_{n}=N_{\varepsilon_{n}}\cap\omega_{1}$, so
$(\delta_{n}:n\in\omega)$ is increasing and
$\delta=\bigcup_{n\in\omega}\delta_{n}$. We define a condition $q$ as follows:

  1. $(\aleph)$

${\rm dom}(q)=N_{\delta}\cap\ell g(K)$.

  2. $(\beth)$

If $\alpha\in{\rm dom}(q)\cap w$ then $q(\alpha)=p_{\delta}(\alpha)$.

  3. $(\gimel)$

If $\alpha\in{\rm dom}(q)-w$ then $q(\alpha)$ is a common upper bound of the
conditions $(q_{\delta_{n}}:n\in\omega)$.

The last item can be instantiated by sweetness. Let
$a=\\{\alpha_{\delta_{n}}:n\in\omega\\}$ and let $S$ be a stationary subset of
$\omega_{1}$ for which $a=A_{\alpha}$ whenever $\alpha\in S$. Here we use the
fact that $(A_{\alpha}:\alpha\in\omega_{1})$ is a diamond sequence in $V$.
Since $q_{\delta_{n}}\leq q$ for every $n\in\omega$ we see that $q\Vdash
a=A_{\alpha}\subseteq\mathchoice{\oalign{$\displaystyle A$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle A$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle A$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle A$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}$ for every $\alpha\in S$, so we are done.

$\qed_{\ref{thmaddm}}$

We can derive now the main result of this section:

######  Corollary 2.3.

Tiltan and ${\rm add}(\mathscr{M})$.
It is consistent that $\clubsuit_{\aleph_{1}}$ holds and ${\rm
add}(\mathscr{M})$ is arbitrarily large.

_Proof_.
Assume $\Diamond_{\aleph_{1}}$ in $V$ and choose an arbitrarily large regular
cardinal $\lambda$ which satisfies $\lambda^{\aleph_{1}}=\lambda$. Let
$K\in\mathcal{K}$ be such that $\lambda=\ell g(K)$ and
$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{K,\beta}$ is Hechler forcing for every $\beta\in\lambda$.
Let $G\subseteq\mathbb{P}_{K}$ be $V$-generic.

By the definition of $\mathcal{K}$ we use a finite support iteration in $K$
and hence Cohen reals are added at limit stages thus ${\rm
cov}(\mathscr{M})=\lambda$ in $V[G]$. By Theorem 2.2 we have
$\mathfrak{b}=\lambda$ in $V[G]$, so ${\rm add}(\mathscr{M})=\lambda$ due to
Miller in [Mil81]. But Theorem 2.2 also ensures $\clubsuit_{\aleph_{1}}$ in
$V[G]$, so the proof is accomplished.

$\qed_{\ref{coraddm}}$

There is an interesting upshot which follows from the above corollary. We
shall phrase and prove it in Corollary 2.5 below, but we need some background.
Let $f$ be a function from $\kappa$ into $\mathcal{P}(\kappa)$. We say that
$f$ is $C(\lambda,\mu)$ iff $|\bigcap\\{f(x):x\in T\\}|<\mu$ for every
$T\in[\kappa]^{\lambda}$. We say that $f$ is $\kappa$-reasonable iff $f$ is
both $C(\kappa,\omega)$ and $C(\omega,\kappa)$. A subset $A$ of $\kappa$ is
$f$-free iff $x\notin f(y)$ whenever $\\{x,y\\}\subseteq A$. Without the
assumption that $f$ is $\kappa$-reasonable, one can find functions
$f:\kappa\rightarrow\mathcal{P}(\kappa)$ for which there are no infinite free
sets (and even no two-element free set, see [Mut15]). However, the mere
assumption of $\kappa$-reasonability is insufficient for getting infinite free
sets, as shown in the above mentioned paper.

It has been proved in [Gar16] that if $f$ is $\kappa$-reasonable and
$\binom{\kappa}{\omega}\rightarrow\binom{\kappa}{\omega}_{2}$ then there
exists an infinite free set for $f$. A natural question is whether this
assumption is necessary. We shall give a negative answer to this question.
Recall that $\binom{\lambda}{\kappa}\rightarrow\binom{\lambda}{\kappa}_{2}$
holds iff for every coloring $c:\lambda\times\kappa\rightarrow 2$ there are
$A\subseteq\lambda$ and $B\subseteq\kappa$ such that $|A|=\lambda,|B|=\kappa$
and $c\upharpoonright(A\times B)$ is constant. Our first statement says that
one can replace the assumption
$\binom{\kappa}{\omega}\rightarrow\binom{\kappa}{\omega}_{2}$ by the weaker
assumption $\binom{\kappa}{\kappa}\rightarrow\binom{\kappa}{\omega}_{2}$.

######  Theorem 2.4.

Infinite free subsets.
Assume that $\binom{\kappa}{\kappa}\rightarrow\binom{\kappa}{\omega}_{2}$ and
$f:\kappa\rightarrow\mathcal{P}(\kappa)$ is $\kappa$-reasonable.
Then there exists an infinite free subset for $f$.

_Proof_.
We define a coloring $c:[\kappa]^{2}\rightarrow 2$ as follows.
$c(\\{\alpha,\beta\\})=1$ iff $\alpha\notin f(\beta)\wedge\beta\notin
f(\alpha)$. We employ the Erdös-Dushnik-Miller theorem to get either
$H_{0}\in[\kappa]^{\kappa}$ such that $c\upharpoonright[H_{0}]^{2}=\\{0\\}$ or
$H_{1}\in[\kappa]^{\omega}$ such that $c\upharpoonright[H_{1}]^{2}=\\{1\\}$.
If there exists such $H_{1}$ then we are done, since it would be a free set
for $f$ by the definition of the coloring $c$, so assume towards contradiction
that there is no $H_{1}$ as above.

It follows that there is a $0$-monochromatic set $H_{0}$ of size $\kappa$. We
decompose it into $H_{0}=A\cup B$ such that $A\cap B=\emptyset$ and
$|A|=|B|=\kappa$. Now we separate the cartesian product $A\times B$ into two
disjoint collections:

$A\times B=\\{\langle a,b\rangle:a\in f(b)\\}\bigcup\\{\langle
a,b\rangle:a\notin f(b)\\}.$

By the assumption
$\binom{\kappa}{\kappa}\rightarrow\binom{\kappa}{\omega}^{1,1}_{2}$ we can
choose $A_{0}\in[A]^{\omega},B_{0}\in[B]^{\kappa}$ such that either
$A_{0}\times B_{0}\subseteq\\{\langle a,b\rangle:a\in f(b)\\}$ or $A_{0}\times
B_{0}\subseteq\\{\langle a,b\rangle:a\notin f(b)\\}$.

If $A_{0}\times B_{0}\subseteq\\{\langle a,b\rangle:a\in f(b)\\}$ then
$A_{0}\subseteq\bigcap\\{f(b):b\in B_{0}\\}$, contradicting the assumption
that $f$ is $C(\kappa,\omega)$. Similarly, if $A_{0}\times
B_{0}\subseteq\\{\langle a,b\rangle:a\notin f(b)\\}$ then $b\in f(a)$ for
every $a\in A_{0},b\in B_{0}$ (since $a\notin f(b)$ and all the members are
taken from $H_{0}$), so $B_{0}\subseteq\bigcap\\{f(a):a\in A_{0}\\}$,
contradicting the assumption that $f$ is $C(\omega,\kappa)$.

$\qed_{\ref{thmtiltanb}}$

Question 2.8 from [Gar16] is whether the assumption
$\binom{\kappa}{\omega}\rightarrow\binom{\kappa}{\omega}_{2}$ is necessary for
proving that every $\kappa$-reasonable function has an infinite free set. In
order to give a negative answer we will show that the positive relation
$\binom{\kappa}{\kappa}\rightarrow\binom{\kappa}{\omega}_{2}$ is consistent
with the negative relation
$\binom{\kappa}{\omega}\nrightarrow\binom{\kappa}{\omega}_{2}$.

######  Corollary 2.5.

Let $\kappa={\rm cf}(\kappa)>\aleph_{1}$.
The following statements are consistent simultaneously:

  1. $(a)$

$2^{\omega}=\kappa$.

  2. $(b)$

$\binom{\kappa}{\omega}\nrightarrow\binom{\kappa}{\omega}_{2}$.

  3. $(c)$

tiltan holds.

  4. $(d)$

every $\kappa$-reasonable function $f:\kappa\rightarrow\mathcal{P}(\kappa)$
has an infinite free set.

_Proof_.
Fix any $\kappa={\rm cf}(\kappa)>\aleph_{1}$. Using Theorem 2.4 we force
tiltan with $\mathfrak{b}=\mathfrak{c}=\kappa$. It follows that
$\mathfrak{b}=\mathfrak{d}=\kappa$ and hence
$\binom{\kappa}{\omega}\nrightarrow\binom{\kappa}{\omega}_{2}$, see Theorem
2.2 of [CGW20]. On the other hand, tiltan implies stick and by Proposition 3.3
of [CGW20] we have
$\binom{\kappa}{\omega_{1}}\rightarrow\binom{\kappa}{\omega}_{2}$ and a
fortiori $\binom{\kappa}{\kappa}\rightarrow\binom{\kappa}{\omega}_{2}$. From
Theorem 2.4 we infer that every $\kappa$-reasonable function
$f:\kappa\rightarrow\mathcal{P}(\kappa)$ has an infinite free set, so we are
done.

$\qed_{\ref{cortiltanfree}}$

We conclude this section with two problems. The first one is about the
relationship between superclub and ${\rm add}(\mathscr{M})$. We do not know
whether club can be replaced by superclub in the main result of this section.
The essential property that we used in the previous section in order to
guarantee the preservation of superclub sequences seems problematic in the
context of Hechler forcing.

######  Question 2.6.

Is it consistent that superclub holds and ${\rm add}(\mathscr{M})>\omega_{1}$?

As mentioned at the beginning of this section, club and hence superclub imply
${\rm add}(\mathscr{N})=\aleph_{1}$, so the results of this section show that
this cannot be stressed further in the horizontal dimension of Cichońs
diagram. Brendle proved in [Bre06] that club is consistent with ${\rm
cov}(\mathscr{N})=\aleph_{2}$, so in some sense we are covered in the vertical
dimension as well. In fact, it follows from the paper of Brendle that tiltan
is consistent with an arbitrarily large value of ${\rm cov}(\mathscr{N})$, see
[Bre06, Section 5].

##  3\. Independent subsets and complete subgraphs

Suppose that $\kappa,\mu\leq\lambda$. The negative square brackets relation
$\lambda\nrightarrow[\kappa]^{2}_{\mu}$ is the assertion that one can find a
coloring $c:[\lambda]^{2}\rightarrow\mu$ such that for every
$A\in[\lambda]^{\kappa}$ one has $c^{\prime\prime}[A]^{2}=\mu$. Namely, every
subset of $\lambda$ of size $\kappa$ assumes all the colors. In the language
of graph theory one can describe this relation as follows. If $G=(V,E)$ is the
complete graph with $\lambda$ vertices and one defines a decomposition
$(E_{\alpha}:\alpha\in\mu)$ of $E$ then every subset $A\subseteq V$ of size
$\kappa$ has at least one edge in every $E_{\alpha}$, that is $[A]^{2}\cap
E_{\alpha}\neq\varnothing$ for every $\alpha\in\mu$.

The fact that $G$ is the complete graph is helpful here, since there are many
edges and hence the probability that each $A$ of size $\kappa$ will intersect
any $E_{\alpha}$ increases. A graph theoretic generalization of this relation
would be the same statement with respect to a larger class of graphs. One has,
however, to insert sufficiently many edges into $G$. Recall that $I\subseteq
V$ is _independent_ iff $I$ is edge-free, that is $[I]^{2}\cap E=\varnothing$.
A necessary requirement for our generalization is the absence of independent
sets of size $\kappa$.

######  Definition 3.1.

$Q(\lambda,\kappa,\mu)$. Assume that $\kappa,\mu\leq\lambda$.
The property $Q(\lambda,\kappa,\mu)$ means that for every graph
$G=(\lambda,E)$ with no independent subset of size $\kappa$ there exists a
partition $(E_{\alpha}:\alpha\in\mu)$ of $E$ such that for every
$\alpha\in\mu$ the graph $G_{\alpha}=(\lambda,E_{\alpha})$ has no independent
subset of size $\kappa$.

The definition of $Q(\lambda,\kappa,\mu)$ comes from [GHK95], and the fact
that it is a strengthening of the square brackets relation with the same
parameters is mentioned there. Let us spell-out the proof.

######  Observation 3.2.

If $Q(\lambda,\kappa,\mu)$ then $\lambda\nrightarrow[\kappa]^{2}_{\mu}$.

_Proof_.
Let $G$ be the complete graph on $\lambda$ vertices and let
$\\{E_{\alpha}:\alpha\in\mu\\}$ be a decomposition of $E$ ensured by the
assumption $Q(\lambda,\kappa,\mu)$. We define a coloring
$c:[\lambda]^{2}\rightarrow\mu$ as follows:

$c(\gamma,\delta)=\alpha\Leftrightarrow\\{\gamma,\delta\\}\in E_{\alpha}.$

Assume that $A\in[\lambda]^{\kappa}$ and $\xi\in\mu$. For every $\alpha\in\mu$
let $G_{\alpha}=(\lambda,E_{\alpha})$. By the assumption
$Q(\lambda,\kappa,\mu)$ we see that $[A]^{2}\cap E_{\alpha}\neq\varnothing$
for every $\alpha\in\mu$, otherwise $A$ would be an independent subset of
$G_{\alpha}$ of size $\kappa$. Fix $\gamma,\delta\in A$ so that
$\\{\gamma,\delta\\}\in E_{\xi}$. By the definition of our coloring we see
that $c(\gamma,\delta)=\xi$. Since $\xi\in\mu$ was arbitrary we conclude that
$c^{\prime\prime}[A]^{2}=\mu$ and the relation
$\lambda\nrightarrow[\kappa]^{2}_{\mu}$ is therefore established.

$\qed_{\ref{obsghk}}$

The property $Q(\lambda,\kappa,\mu)$ is strictly stronger than the relation
$\lambda\nrightarrow[\kappa]^{2}_{\mu}$. It has been proved in [GHK95] that
the negation of $Q(\aleph_{1},\aleph_{1},\aleph_{1})$ is consistent.
Contrariwise, Todorčević proved in [Tod87] that
$\aleph_{1}\nrightarrow[\aleph_{1}]^{2}_{\aleph_{1}}$ holds in ZFC. Hence the
implication in the above observation cannot be reversed. However, for the
negation of $Q(\aleph_{1},\aleph_{1},\aleph_{1})$ one has to violate CH. More
generally, it is shown in [GHK95] that if $2^{\kappa}=\kappa^{+}$ then
$Q(\kappa^{+},\kappa^{+},\kappa^{+})$ holds. Let us prove that this conclusion
follows from a weaker assumption. Namely, rather than the strong assumption
$2^{\kappa}=\kappa^{+}$ one can assume the weaker assumption that superclub
holds at $\kappa^{+}$ and still derive $Q(\kappa^{+},\kappa^{+},\kappa^{+})$.

######  Theorem 3.3.

Let $\kappa$ be a regular cardinal. If superclub holds at $\kappa^{+}$ then
$Q(\kappa^{+},\kappa^{+},\kappa^{+})$ holds.

_Proof_.
Suppose that $G=(\kappa^{+},E)$ contains no independent subset of size
$\kappa^{+}$. Let $S$ be $S^{\kappa^{+}}_{\kappa}$, and let superclub be
exemplified by $(S_{\alpha}:\alpha\in S)$. Set $S_{\alpha}=\varnothing$ for
every $\alpha\in\kappa^{+}-S$. We would like to define a function
$f:E\rightarrow\kappa^{+}$ with the following two properties at every
$\alpha\in\kappa^{+}$:

  1. $(a)$

For every $\\{\beta,\alpha\\}\in E$ where
$\beta<\alpha,f(\beta,\alpha)<\alpha$.

  2. $(b)$

If $\gamma<\alpha$ and $|\\{\beta\in S_{\gamma}:\\{\beta,\alpha\\}\in
E|=\kappa$ then $\\{f(\beta,\alpha):\beta\in
S_{\gamma}\wedge\\{\beta,\alpha\\}\in E\\}=\alpha$.

In order to define $f$ notice that if $\alpha\in\kappa^{+}$ then the
cardinality of the set $x_{\alpha}=\\{\gamma\in\alpha:|\\{\beta\in
S_{\gamma}:\\{\beta,\alpha\\}\in E|=\kappa\\}$ is at most
$|\alpha|\leq\kappa$. So we choose a disjoint refinement
$\\{T_{\gamma}:\gamma\in x_{\alpha}\\}$ for the collection
$\\{S_{\gamma}:\gamma\in x_{\alpha}\\}$ and we fix a bijection
$h^{\alpha}_{\gamma}:T_{\gamma}\rightarrow\alpha$. For every $\gamma\in
x_{\alpha}$ and every $\beta\in T_{\gamma}$ we let
$f(\beta,\alpha)=h^{\alpha}_{\gamma}(\beta)$ and we define $f(\beta,\alpha)=0$
otherwise.

Having defined the function $f$ we can depict our decomposition of $E$. For
every $\tau\in\kappa^{+}$ set $\\{\beta,\alpha\\}\in E_{\tau}\Leftrightarrow
f(\beta,\alpha)=\tau$. Notice that $\\{E_{\tau}:\tau\in\kappa^{+}\\}$ is a
decomposition of $E$ and we claim that it exemplifies
$Q(\kappa^{+},\kappa^{+},\kappa^{+})$ with respect to $G=(\kappa^{+},E)$.

To see this, suppose that $H\subseteq\kappa^{+}$ and $|H|=\kappa^{+}$. By
induction on $\delta\in\kappa^{+}$ we choose an element $y_{\delta}\in H$ as
follows. As a first step, let $\\{S_{\gamma_{i}}:i\in\kappa^{+}\\}$ be a
subset of the superclub sequence with the following two properties:

  1. $(a)$

$S_{\gamma_{i}}\subseteq H$ for every $i\in\kappa^{+}$.

  2. $(b)$

If $i<j<\kappa^{+}$ then $S_{\gamma_{j}}$ end-extends $S_{\gamma_{i}}$.

Such a collection exists since one can choose $I\subseteq H,|I|=\kappa^{+}$ so
that $S_{I}=\\{\gamma\in\kappa^{+}:I\cap\gamma=S_{\gamma}\\}$ is a stationary
subset of $\kappa^{+}$. Enumerate the elements of $S_{I}$ by
$\\{S_{\gamma_{i}}:i\in\kappa^{+}\\}$ and verify the above properties.

Our second step is rendered by induction on $\delta\in\kappa^{+}$. Suppose
that $A_{\delta}=\\{y_{\zeta}:\zeta\in\delta\\}$ has been chosen and we try to
choose $y_{\delta}$. We will assume that $A_{\delta}\subseteq S_{\gamma_{i}}$
for some $i\in\kappa^{+}$, and let $i(\delta)$ be the first such ordinal.
Choose $y_{\delta}\in I\subseteq H$ so that $y_{\delta}>\gamma_{i(\delta)}$
and hence, in particular, $y_{\delta}>y_{\zeta}$ for every $\zeta\in\delta$
since $A_{\delta}\subseteq S_{\gamma_{i(\delta)}}\subseteq\gamma_{i(\delta)}$.
Notice that $A_{\delta+1}\subseteq S_{\gamma_{j}}$ for some $j\in\kappa^{+}$.

By way of contradiction assume that $\xi\in\kappa^{+}$ and $H$ is an
independent subset of the graph $(\kappa^{+},E_{\xi})$. This means that
$\xi\notin f^{\prime\prime}[H]^{2}$. By the construction, if $\delta>\xi$ then
the set $z_{\delta}=\\{\beta\in\delta:\\{y_{\beta},y_{\delta}\\}\in E\\}$ is
of size less than $\kappa$ since otherwise the image of $f$ on these pairs
will cover all the ordinals of $\delta$ including $\xi$.

We employ now the free subset theorem, which says (in this case) that if
$h:\kappa^{+}\rightarrow\mathcal{P}(\kappa^{+})$ satisfies
$|h(\alpha)|<\kappa$ for every $\alpha\in\kappa^{+}$ then there exists an
$h$-free subset $F\subseteq\kappa^{+}$ of size $\kappa^{+}$. Consider the set-
mapping $g:\kappa^{+}\rightarrow\mathcal{P}(\kappa^{+})$ given by
$g(\delta)=z_{\delta}$ for every $\delta\in\kappa^{+}$. By the free subset
theorem there is $F\subseteq\kappa^{+},|F|=\kappa^{+}$ which is $g$-free.
Namely, if $\delta,\eta\in F$ and $\delta<\eta$ then $\delta\notin z_{\eta}$
and hence $\\{y_{\delta},y_{\eta}\\}\notin E$. It follows that
$\\{y_{\beta}:\beta\in F\\}$ is an independent subset of $G=(\kappa^{+},E)$, a
contradiction.

$\qed_{\ref{thmsuperclub}}$

We do not know whether $\clubsuit_{\kappa^{+}}$ is consistent with the failure
of the principle $Q(\kappa^{+},\kappa^{+},\kappa^{+})$. It is clear, however,
that the strategy of [FSS97] cannot yield such a result, since it cannot
separate club from superclub. On the other hand, the method of [She98] is
promising in this context. The idea is to force some property at $\aleph_{2}$
while keeping $\clubsuit_{S}$ at $S=S^{\omega_{2}}_{\omega}$ and then to
collapse $\aleph_{1}$. The collapse secures the club principle at $\omega_{1}$
but destroys superclub. This method is limited to properties which can be
forced at $\aleph_{2}$ and one also has to make sure that the pertinent
property is preserved by the collapse.

######  Question 3.4.

Is it consistent that $\clubsuit_{\kappa^{+}}$ holds but
$Q(\kappa^{+},\kappa^{+},\kappa^{+})$ fails for some infinite cardinal
$\kappa$?

Let us indicate that the only property which separates tiltan from superclub,
as far as we know, is Galvin’s property. Thus, a positive answer to the above
question would be interesting.

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