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

We investigate the function LH(n)=max⁡{|H∩P(A)|:|A|=n}LH(n)=max⁡{|H∩P(A)|:|A|=n} where HH is a set of finite subsets of λ such that every λ-sized subset of λ   has arbitrarily large subsets form HH. For κ=ℵ1κ=ℵ1, lim⁡LH(n)/n→∞lim⁡LH(n)/n→∞ and LH(n)=O(n2)LH(n)=O(n2), and in different models of set theory, either bound can be sharp. If λ>ℵ1λ>ℵ1, LH(n)>cn2LH(n)>cn2 for some c>0c>0 and n sufficiently large. If λ   is strong limit singular, then LHLH is superpolynomial. If κ<λκ<λ are uncountable cardinals, we call a family HHκ-dense (strongly κ-dense) in λ if every κ-sized subset of λ   contains a set (arbitrarily large sets) in HH. We show under GCH that if HH is κ+κ+-dense in κ+rκ+r (r   finite), then LH(n)/nr→∞LH(n)/nr→∞ (κ=ωκ=ω) and LH(n)>cnr+1LH(n)>cnr+1 (κ>ωκ>ω). We also give bounds for LH(n)LH(n) when HH has large chromatic or coloring number.