A problem of Laczkovich: How dense are set systems with no large independent sets?

Abstract

We investigate the function LH(n)=max⁡{|H∩P(A)|:|A|=n} where H is a set of finite subsets of λ such that every λ-sized subset of λ has arbitrarily large subsets form H. For κ=ℵ1, lim⁡LH(n)/n→∞ and LH(n)=O(n2), and in different models of set theory, either bound can be sharp. If λ>ℵ1, LH(n)>cn2 for some c>0 and n sufficiently large. If λ is strong limit singular, then LH is superpolynomial. If κ<λ are uncountable cardinals, we call a family Hκ-dense (strongly κ-dense) in λ if every κ-sized subset of λ contains a set (arbitrarily large sets) in H. We show under GCH that if H is κ+-dense in κ+r (r finite), then LH(n)/nr→∞ (κ=ω) and LH(n)>cnr+1 (κ>ω). We also give bounds for LH(n) when H has large chromatic or coloring number.