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Are there infinite cardinals $\kappa < \lambda$ such that here is a collection ${\cal A}$ of subsets of $\lambda$ with the following properties:

  1. $|{\cal A}| = 2^\lambda$, and

  2. $A\neq B\in {\cal A}$ implies $|A\cap B|\leq \kappa$.

Dominic van der Zypen
  • 45,374

1 Answers1

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Here's a quick observation: suppose $2^\kappa=\kappa^+$. Then we can mimic the construction of an almost disjoint set of sets of natural numbers of size continuum: to each map $p: \kappa^+\rightarrow 2$, we associate the set of small approximations $A_p=\{p\upharpoonright\alpha:\alpha<\kappa^+\}$. By $2^\kappa=\kappa^+$ we have a bijection from partial maps from $\kappa^+$ to $2$ with bounded domain, and $\kappa^+$ itself; so we may replace each $A_p$ with a corresponding $B_p\subseteq \kappa^+$. The collection $\{B_p: p\in 2^{\kappa^+}\}$ is a collection of subsets of subsets of $\kappa^+$, with cardinality $2^{\kappa^+}$, with pairwise intersections of sizes $\le\kappa$.

So in order to have any hope for your principle to fail, we would need to work in a universe where GCH fails everywhere; and the global failure of GCH has large cardinal strength.

Noah Schweber
  • 18,932
  • I thought a $\Delta$-system was a family of sets whose pairwise intersections were identical. Did the nomenclature change, or was I always wrong about that? – bof Aug 09 '17 at 03:22
  • @bof The body of the question does not use the word "$\Delta$-system," and the way the second condition is phrased makes it sound very not $\Delta$-systemy; I'm taking the body over the title here. The OP of course can indicate whether I'm actually addressing their question. – Noah Schweber Aug 09 '17 at 03:59
  • Thanks @NoahSchweber! 1) I would like to wait for a while till I accept the answer to see whether somebody comes up constructing ${\cal A}$ with the disired properties within ${\sf ZFC}$; 2)Re: Terminology, I apologize if I used $\Delta$-system in a wrong way, but I'm reluctant to change the title. – Dominic van der Zypen Aug 09 '17 at 11:53
  • @DominicvanderZypen Can you clarify what is the actual connection between this question and $\Delta$-systems? – Noah Schweber Aug 29 '17 at 17:22
  • Oh - I thought the set system I defined in the question is referred to as "$\Delta$-system". What is the correct term? - I'll change the title to "Set systems [...]" – Dominic van der Zypen Aug 30 '17 at 07:05
  • @DominicvanderZypen I don't know if there's a specific term for what you're asking about, but a $\Delta$-system is a collection of sets with the same intersection, not small intersection; see here. – Noah Schweber Sep 11 '17 at 23:10
  • @NoahSchweber Thanks for your comment, and sorry for creating some confusion by my wrong use of $\Delta$-system – Dominic van der Zypen Sep 12 '17 at 13:23