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Let $\Delta(\kappa, \mu)$ be the statement: "let $F$ be a family of cardinality $\kappa$ of sets of cardinality less than $\mu$. Then there is a family $G \subset F$ of cardinality $\kappa$ and a set $r$ such that $a \cap b=r$ for every $a,b \in G$".

We know that if $\kappa$ is a regular cardinal and $\lambda^{<\mu} < \kappa$, for every $\lambda < \kappa$ then $\Delta(\kappa, \mu)$ holds. My question is:

Does $\Delta(\kappa, \mu)$ for some regular uncountable $\kappa$ and some uncountable $\mu<\kappa$ imply some weak form of the GCH below $\kappa$?

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Do you really mean to ask about the GCH at $\kappa$, rather than, say, at the cardinal predecessors of $\kappa$ in the case that $\kappa$ is a successor? For example, in the positive direction your hypothesis is not really a GCH property at $\kappa$ itself, but a GCH property below $\kappa$. –  Joel David Hamkins May 17 '13 at 0:40
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Santi, I like this question. I have wondered what assumptions we make when using a delta-system argument. –  Erin Carmody May 17 '13 at 7:17
    
Yes, Joel, I did mean "below $\kappa$". –  Santi Spadaro May 17 '13 at 14:28

1 Answer 1

up vote 15 down vote accepted

It is a very nice question! The answer is yes, natural instances of the $\Delta$ system property, which hold under GCH, are in fact equivalent to the GCH.

Theorem. $\Delta(\omega_2,\omega_1)$ is equivalent to CH.

Proof: You've pointed out that CH implies the principle, since the hypothesis you mention for this case amounts to $\omega_1^{\lt\omega_1}<\omega_2$, which amounts to CH. So let us consider what happens when CH fails. Let $T=2^{\lt\omega}$ be the tree of all finite binary sequences, and label the nodes of $T$ with distinct natural numbers. Let $F$ be the subsets of $\omega$ arising as the sets of labels occuring on any of $\omega_2$ many branches through $T$. Thus, $F$ has size $\omega_2$, and any two elements of $F$ have finite intersection. I claim that this family of sets can have no $\Delta$-system of size $\omega_2$, and indeed, it can have no $\Delta$-system even with three elements. If $r$ is the root of $a$, $b$ and $c$ in $F$, then $r=a\cap b=a\cap c$, and so $a$ and $b$ branch out at the same node that $a$ and $c$ do, in which case $b$ and $c$ must agree one step longer, so $b\cap c\neq r$. QED

The same idea works for higher cardinals as follows:

Theorem. For any infinite cardinal $\delta$, we have $\Delta(\delta^{++},\delta^+)$ is equivalent to $2^\delta=\delta^+$.

Proof. If $2^\delta=\delta^+$, then your criterion, which amounts to $(\delta^+)^{\lt\delta^+}<\delta^{++}$, is fulfilled, and so the $\Delta$ property holds. Conversely, consider the tree $T=2^{\lt\delta}$, the binary sequences of length less than $\delta$. Let $F$ be a family of $\delta^{++}$ many branches through $T$, regarding each branch $b$ as a subset of $T$, the set of its initial segments. Each such branch has size $\delta$, since the tree has height $\delta$. But for the same reason as before, there can be no $\Delta$ system with even three elements, since the tree is merely binary branching, and so three distinct branches cannot have a common root. This contradicts $\Delta(\delta^{++},\delta^+)$, as desired. QED

Corollary. The full GCH is equivalent to the assertion that $\Delta(\delta^{++},\delta^+)$ for every infinite cardinal $\delta$.


Update. The same idea shows that the hypothesis you mention is optimal: one can reverse the lemma from the conclusion to the hypothesis.

Theorem. The following are equivalent, for regular $\kappa$ and $\mu\lt\kappa$:

  1. $\Delta(\kappa,\mu)$
  2. $\lambda^{\lt\mu}\lt\kappa$ for every $\lambda\lt\kappa$.

Proof. You mentioned that 2 implies 1, and this is how one usually sees the $\Delta$ system lemma stated. For the converse, suppose that $\lambda^{\lt\mu}\geq\kappa$ for some $\lambda\lt\kappa$. Since $\kappa$ is regular and $\mu\lt\kappa$, this implies $\lambda^\eta\geq\kappa$ for some $\eta\lt\mu$. Let $T$ be the $\lambda$-branching tree $\lambda^{\lt\eta}$, which has height $\eta$. Let $F$ be a family of $\kappa$ many branches through this tree, where we think of a branch as the set of nodes in the tree that lie on it, a maximal linearly ordered subset of the tree $T$. Each such branch is a set of size $\eta$. I claim that this family has no subfamily that is $\Delta$ system of size $\lambda^+$. The reason is that because the tree is $\lambda$-branching, if we have $\lambda^+$ many branches with a common root, then at least two of them must extend that root to the next level in the same way, a contradiction to it being a root. Thus, the failure of 2 implies the failure of 1, as desired. QED

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It is interesting! What happens when $\Delta( \delta ^{+++++++}, \delta^+)$ ? –  Erin Carmody May 17 '13 at 7:22
    
Erin - $\Delta( \delta ^{+++++++}, \delta^+) => \Delta( \delta ^{++}, \delta^+)$ –  Eran May 17 '13 at 10:00
    
@Joel - isn't this proof goes just as well assuming ZFC + GCH + KH (Kurepa's Hypothesis)? –  Eran May 17 '13 at 10:07
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Very nice, Joel! It's more than I hoped. –  Santi Spadaro May 17 '13 at 14:27
    
Erin, my update shows that $\Delta(\delta^{+++++++},\delta^+)$ is equivalent to the assertion that $(\delta^{++++++})^\delta=\delta^{++++++}$, which is equivalent to saying $2^\delta$ is at most $\delta^{++++++}$, a weak form of the GCH. –  Joel David Hamkins May 17 '13 at 17:39

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