On extremal problems concerning the traces of sets

Given two non-negative integers n and s, define m(n, s) to be the maximal number m such that every hypergraph ℋ on n vertices and with at most m edges has a vertex x such that |ℋ_x|≥|E(ℋ)|−s, where ℋ_x={H∖{x}:H∈E(ℋx)}. The problem of determining the limit m(s)=lim_n→∞m(n,s)/n was posed by Füredi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of s, Frankl determined m(2^d−1−1) for all d∈ℕ. Here we prove that m(2^d−1−c)=(2^d−c)/d for every c,d∈ℕ with d≥4c and give an example showing that this equality does not hold anymore for d=c.

The other line of research on this problem is to determine m(s) for small values of s. In this line, our second result determines m(2^d−1−c) for c∈{3,4}. This solves more instances of the problem for small s and in particular solves a conjecture by Frankl and Watanabe.