Sidorenko's conjecture

Theorem 1.1: for the explicit bipartite incidence graph $H$ of 22 triples on 13 points ($35$ vertices, $66$ edges, connected, degrees 4 to 6 on one side and 3 on the other) there is a finite simple graph $G$ with an edge and $t(H,G)<p(G)^{66}$. Corollaries: tensor powers make $t(H,F)/p(F)^{66}$ arbitrarily small, and blow-ups give a fixed deficit for injective copies at a positive limiting density. Not given: an explicit host graph, the smallest counterexample, or which other bipartite graphs fail.

Theorem 1.1: for the explicit bipartite incidence graph $H$ of 22 triples on 13 points ($35$ vertices, $66$ edges, connected, degrees 4 to 6 on one side and 3 on the other) there is a finite simple graph $G$ with an edge and $t(H,G)<p(G)^{66}$. Corollaries: tensor powers make $t(H,F)/p(F)^{66}$ arbitrarily small, and blow-ups give a fixed deficit for injective copies at a positive limiting density. Not given: an explicit host graph, the smallest counterexample, or which other bipartite graphs fail.