Triangle-Intersecting Families of Graphs

TL;DR

This paper proves a longstanding conjecture that the largest triangle-intersecting families of graphs are those containing a fixed triangle, and extends the result to odd-cycle-intersecting families with stability analysis.

Contribution

It proves the 1976 conjecture by Simonovits and Sos and generalizes the result to odd-cycle-intersecting families, providing bounds and stability results.

Findings

Confirmed the maximum size of triangle-intersecting families as (1/8) 2^{n choose 2}

Extended the result to odd-cycle-intersecting families

Established stability results for near-largest families

Abstract

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.