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 $G_1,G_2 \in F$, the intersection $G_1 \cap G_2$ 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 sub-graph isomorphic to $H$. In [D. Ellis, Y. Filmus, and E. Friedgut, Triangle-intersecting families of graphs, J. Eur. Math. Soc. 14 (2012), pp. 841--885], Ellis, Filmus and Friedgut proved a 36-year old conjecture of Simonovits and S\'{o}s stating that the maximal size of a triangle-intersecting family is $(1/8)2^{n(n-1)/2}$. Furthermore, they proved a $p$-biased generalization, stating that for any $p \leq 1/2$, we have $\mu_{p}(F)\le p^{3}$, where $\mu_{p}(F)$ is the probability that the random graph $G(n,p)$ belongs to $F$. In the same paper, Ellis et al. conjectured that the assertion of their biased theorem holds also for $1/2 < p \le 3/4$, and more generally, that for any non-$t$-colorable graph $H$ and any $H$-intersecting family $F$, we have $\mu_{p}(F)\le p^{t(t+1)/2}$ for all $p \leq (2t-1)/(2t)$. In this note we construct, for any fixed $H$ and any $p>1/2$, an $H$-intersecting family $F$ of graphs such that $\mu_{p}(F)\ge 1-e^{-n^{2}/C}$, where $C$ depends only on $H$ and $p$, thus disproving both conjectures.