A note on large H-intersecting families

A family F of graphs on a fixed set of n vertices is called triangle-intersecting if for any G1,G2 ∈ F , the intersection G1 ∩ G2 contains a triangle. More generally, for a fixed graph H, a family F is H-intersecting if the intersection of any two graphs in F contains a subgraph isomorphic to H. In [D. Ellis, Y. Filmus, and E. Friedgut, J. Eur. Math. Soc., 14 (2012), pp. 841-885], a 36-year old conjecture of Simonovits and Sós was proved stating that the maximal size of a triangleintersecting family is (1/8)2ⁿ (n-1)/2. Furthermore, they proved a p-biased generalization, stating that for any p ≤ 1/2, we have μ_p (F ) ≤ p³, where μ p (F ) is the probability that the random graph G(n, p) belongs to \scrF . In the same paper, the authors conjectured that the assertion of their biased theorem holds also for 1/2 < p ≤ 3/4, and more generally, that for any non-t-colorable graph H and any H-intersecting family F , we have μ p (F ) ≤ p^t (t+1)/2 for all p ≤ (2t 1)/(2t). In this note we construct, for any fixed H and any p > 1/2, an H-intersecting family F of graphs such that μ p (F ) ≥ 1 e^-n2/C, where C depends only on H and p, thus disproving both conjectures.