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arXiv · 1010.4909

Triangle-Intersecting Families of Graphs

David Ellis·Yuval Filmus·Ehud Friedgut
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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.

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David Ellis, Yuval Filmus, Ehud Friedgut. 2010-10-23. Triangle-Intersecting Families of Graphs. https://arxiv.org/abs/1010.4909

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