Triangle-intersecting families of graphs

David Ellis, Yuval Filmus and Ehud Friedgut
Journal of the European Mathematical Society, Volume 14, Issue 3, 2012, pp. 841–885

A family of graphs is said to be triangle-intersecting if the intersection of any two graphs in the family contains a triangle. A conjecture of Simonovits and Sós 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 containing 1/8 of all graphs.. 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.

The arXiv version corrects a mistake in the proof of the stability part of Corollary 2.3. The original proof assumed that the approximating family $G$ is odd-cycle-agreeing, and deduced that $G$ must be a triangle-junta from uniqueness. However, showing that $G$ is odd-cycle-agreeing requires a separate argument, found in the new version.

An alternative presentation of the results in this paper can be found in my PhD thesis.


 author = {David Ellis and Yuval Filmus and Ehud Friedgut},
 title = {Triangle-intersecting families of graphs},
 journal = {Journal of the European Mathematical Society},
 volume = {14},
 number = {3},
 year = {2012},
 pages = {841--885}
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