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 such that in every hypergraph H on n vertices and with at most m(n, s) edges there is a vertex x such that vertical bar H-x vertical bar >= vertical bar E(H)vertical bar - s, where H-x = {H \ {x} : H is an element of E(H)}. This problem has been posed by Furedi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of s, Frankl determined m(n, 2(d-1) - 1) for all d subset of N with d vertical bar n. Subsequently, the goal became to determine (n, 2(d-1) - c) for larger c. Frankl and Watanabe determined (n, 2(d-1) - c) for c is an element of {0, 2}. Other general results were not known so far. Our main result sheds light on what happens further away from powers of two: We prove that (n, 2(d-1) - c) = n/d(2(d) - c) for d >= 4c and d vertical bar n and give an example showing that this equality does not hold for c = d. The other line of research on this problem is to determine m(n, s) for small values of s. In this line, our second result determines (n, 2(d-1) - c) for c is an element of {3, 4}. This solves more instances of the problem for small s and in particular solves a conjecture by Frankl and Watanabe. (C) 2021 Elsevier Inc. All rights reserved.