On extremal problems concerning the traces of sets
Abstract
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 |Hx|≥ | E(H)| -s, where Hx=\H\x\:H∈ E(H)\. This problem has been posed by F\"uredi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of s, Frankl determined m(n,2d-1-1) for all d∈N with d n. Subsequently, the goal became to determine m(n,2d-1-c) for larger c. Frankl and Watanabe determined m(n,2d-1-c) for c∈\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 m(n,2d-1-c)=nd(2d-c) for d≥ 4c and d 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 m(n,2d-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.