Four-vertex traces of finite sets

TL;DR

This paper investigates the maximum size of set families with restricted traces on 4-element subsets, establishing bounds and conditions for when larger families must have larger traces on some 4-set.

Contribution

It introduces new bounds on set family sizes with trace restrictions on 4-sets, extending combinatorial understanding of set systems.

Findings

For large n, sets larger than a specific family must have a trace of at least 13 on some 4-set.

Established bounds for traces of set families on 4-element subsets.

Proved that certain size conditions force larger traces on some 4-set.

Abstract

Let $[n]=X_1\cup X_2\cup X_3$ be a partition with $\lfloor\frac{n}{3}\rfloor \leq |X_i|\leq \lceil\frac{n}{3}\rceil$ and define $\mathcal{G}=\{G\subset [n]\colon |G\cap X_i|\leq 1, 1\leq i\leq 3\}$. It is easy to check that the trace $\mathcal{G}_{\mid Y}:=\{G\cap Y\colon G\in \mathcal{G}\}$ satisfies $|\mathcal{G}_{\mid Y}|\leq 12$ for all 4-sets $Y\subset [n]$. For $n\geq 25$ it is proven that whenever $\mathcal{F}\subset 2^{[n]}$ satisfies $|\mathcal{F}|>|\mathcal{G}|$ then $|\mathcal{F}_{\mid C}|\geq 13$ for some $C\subset [n]$, $|C|=4$. Several further results of a similar flavor are established as well.