Cross-free families have linear size

TL;DR

This paper proves that families of subsets with no k-pairwise crossing members on an n-element set have size linearly bounded by n, resolving a longstanding conjecture in combinatorics.

Contribution

It establishes an upper bound of O_k(n) for the size of such families, confirming the conjecture by Karzanov and Lomonosov.

Findings

Families with no k-pairwise crossing subsets are linearly bounded in size.

The proven bound is tight up to constant factors depending on k.

This result settles a major open problem in the growth rate of crossing-free families.

Abstract

Two subsets $A$ and $B$ of a ground set $X$ are \emph{crossing} if none of the four sets $A\setminus B,B\setminus A,A\cap B, X\setminus (A\cup B)$ are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an $n$-element ground set with no $k$-pairwise crossing members has size $O(kn)$. We prove the bound $O_k(n)$, settling (arguably) the main problem about the growth rate of such families.