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Cross-free families have linear size

István Tomon

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.

Cross-free families have linear size

Abstract

Two subsets and of a ground set are \emph{crossing} if none of the four sets are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an -element ground set with no -pairwise crossing members has size . We prove the bound , settling (arguably) the main problem about the growth rate of such families.
Paper Structure (3 sections, 11 theorems, 18 equations, 1 figure)

This paper contains 3 sections, 11 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weakly-cross-free families

Key Result

Theorem 1.1

For every $k$, there exists $c_k>0$ such that the following holds. Let $X$ be a set of size $n$, and let $\mathcal{F}\subset 2^X$ be a $k$-cross-free family. Then $|\mathcal{F}|\leq c_kn$.

Figures (1)

  • Figure 1: An example of a cross-support tree. Each node represents a chain of size 3. We use the shorthand 123 for $\{1,2,3\}$. The sets $S_v$ are written in bold, and the red edges highlight the containment between the sets $S_v$ required by T5.

Theorems & Definitions (23)

  • Theorem 1.1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 3.1
  • Lemma 5
  • ...and 13 more