@inproceedings{205ad163b0274623af2e341ac8dcf443,

title = "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 m such that every hypergraph ℋ on n vertices and with at most m edges has a vertex x such that |ℋx|≥|E(ℋ)|−s, where ℋx={H∖{x}:H∈E(ℋx)}. The problem of determining the limit m(s)=limn→∞m(n,s)/n was posed by F{\"u}redi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of s, Frankl determined m(2d−1−1) for all d∈ℕ. Here we prove that m(2d−1−c)=(2d−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(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.",

keywords = "Extremal set theory, Traces of sets, Abstract simplicial complexes",

author = "S. Piga and B. Sch{\"u}lke",

year = "2021",

month = aug,

day = "24",

doi = "10.1007/978-3-030-83823-2_104",

language = "English",

isbn = "9783030838225",

series = "Trends in Mathematics",

publisher = "Birkhauser",

pages = "651--656",

editor = "Jaroslav Ne{\v s}et{\v r}il and Guillem Perarnau and Juanjo Ru{\'e} and Oriol Serra",

booktitle = "Extended Abstracts EuroComb 2021",

edition = "1",

}