Traces Without Maximal Chains

TL;DR

This paper proves Patkós' conjecture that for large n, the maximum size of a family of k-sets with traces on r-subsets avoiding maximal chains is bounded by a specific binomial coefficient.

Contribution

The paper confirms Patkós' conjecture, establishing an exact upper bound on the size of such set families for large n.

Findings

Proves Patkós' conjecture for large n.

Establishes the maximum size of set families avoiding maximal chains in traces.

Provides combinatorial bounds related to set family traces.

Abstract

The trace of a family of sets $\mathcal{A}$ on a set $X$ is $\mathcal{A}|_X=\{A\cap X:A\in \mathcal{A}\}$. If $\mathcal{A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace $\mathcal{A}|_X$ does not contain a maximal chain, then how large can $\mathcal{A}$ be? Patk\'os conjectured that, for $n$ sufficiently large, the size of $\mathcal{A}$ is at most $\binom{n-k+r-1}{r-1}$. Our aim in this paper is to prove this conjecture.