Ellis–Filmus–Friedgut conjecture for clique-intersecting graph families when t is at least 5

Prove that every family of graphs on a common vertex set of n labeled vertices in which every two graphs share a copy of the complete graph K_t has size at most 2^{\binom{n}{2}-\binom{t}{2}}, with equality attained by the family of all graphs containing a fixed K_t, for every integer t\geq 5.

Background

The paper recalls the conjecture that the extremal K_t-intersecting family consists of all graphs containing one fixed t-clique. The conjecture is known in the paper’s discussion for t=2 and t=3, and a result of Berger and Zhao is cited for t=4. The stated unresolved range is therefore t\geq 5.

The paper’s entropy argument establishes a weaker upper bound, namely 2^{\binom{n}{2}-(t-1)}, for K_t-intersecting families. Thus, resolving the open case would require attaining the substantially sharper exponent predicted by the conjecture.