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Does the following hold?

For every bipartite graph $H$ and every graph $G$ with $e(G)\geq 0.1(v(G))^2$, $$t(H,G)\geq t(K_2, G)^{e(H)}.$$

If not sure, is this a equal question as Sidorenko's conjecture or a subquestion of Sidorenko?

$t(H,G)$ means: Take a random onto map $f$ from $V(H) \to V(G)$, the prob. that this map keep adjacency, i.e. a~b implies f(a)~f(b). If G's adjacency matrix's every element is 1, then for every $H$, $t(H,G)=V(G)^{V(H)}/V(G)^{V(H)}=1$.

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  • $\begingroup$ Please define your terms. What is $t$? $\endgroup$
    – Brendan McKay
    Commented Sep 22 at 11:38
  • $\begingroup$ @BrendanMcKay Take a random onto map f from V(H) to V(G), the prob. that this map keep adjacency, i.e. a~b implies f(a)~f(b). If G's adjacency matrix's every element is 1, then for every H t(H,G)=V(G)^V(H)/V(G)^V(H)=1. $\endgroup$
    – tom jerry
    Commented Sep 22 at 14:44
  • $\begingroup$ @BrendanMcKay $t$ denotes homomorphism density: en.m.wikipedia.org/wiki/Homomorphism_density $\endgroup$
    – TheBestMagician
    Commented Sep 22 at 22:43

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