Triangle-intersecting families on eight vertices

TL;DR

This paper presents an elementary proof for the maximum size of triangle-intersecting graph families on eight vertices, confirming a special case of a longstanding conjecture.

Contribution

It offers a new, elementary proof for the case of eight vertices, simplifying previous spectral method approaches.

Findings

Confirmed the maximum size for triangle-intersecting families on 8 vertices

Provided an elementary proof using partition arguments

Validated a special case of Simonovits and Sós conjecture

Abstract

Simonovits and S\'{o}s conjectured that the maximal size of a triangle-intersecting family of graphs on $n$ vertices is $2^{\binom{n}{2}-3}$. Their conjecture has recently been proved using spectral methods. We provide an elementary proof of the special case of $8$ vertices using a partition argument.