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 $G_1 \cap G_2$containsatriangle\mathrm{.}Moregenerally,forafixedgraph$H$,afamily$F$is$H$-intersectingiftheintersectionofanytwographsin$F$ 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 $(1\mathrm{/}8)2^{n(n-1)\mathrm{/}2}$. Furthermore, they proved a $p$-biased generalization, stating that for any $p \leq 1/2$,wehave$\mu_{p}(F)\le p^{3}$,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 $1\mathrm{/}2<p\le 3\mathrm{/}4$, and more generally, that for any non-$t$-colorable graph $H$ and any $H$-intersecting family $F$, we have ${\mu }_p(F)\le p^{t(t+1)\mathrm{/}2}$ for all $p\le (2t-1)\mathrm{/}(2t)$. In this note we construct, for any fixed $H$ and any $p>1\mathrm{/}2$, an $H$-intersecting family $F$ of graphs such that ${\mu }_p(F)\ge 1-e^{-n^2\mathrm{/}C}$, where $C$ depends only on $H$ and $p$, thus disproving both conjectures.