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  "body": "\n## Statement\n\nLet $H$ and $G$ be finite simple [[graphs|00C0BR7D7H]], with $G$ having at least one vertex.\nLet $\\hom(H, G)$ be the number of [[graph homomorphisms|052JJ9YVNM]] from $H$ to $G$, and let\n$$t(H, G) = \\frac{\\hom(H, G)}{|V(G)|^{|V(H)|}}$$\nbe the homomorphism density.\nLet $K_2$ be the graph with one edge, so that $t(K_2, G) = 2 |E(G)| / |V(G)|^2$.\nLet $H$ be a [[bipartite graph|0ZVQKTRKVQ]] with $e(H)$ edges.\nThen for every graph $G$,\n$$t(H, G) \\ge t(K_2, G)^{e(H)}.$$\n\n## Remarks\n\nThe conjecture is open.\nIt is known for trees, for even cycles, for complete bipartite graphs and for hypercube graphs.\nIt is also known for every bipartite graph that has a vertex adjacent to all vertices of the other part.\n\nThe right side is the density of $H$ in a random graph with the same edge density.\nSo the conjecture says that random graphs minimise the density of $H$ among graphs of a given edge density.\nBipartiteness is necessary.\nIf $H$ has an odd cycle and $G$ is the complete bipartite graph $K_{n,n}$, then $t(H, G) = 0$, while $t(K_2, G) = 1/2$.\nFor $H$ a path with two edges, the conjecture is the inequality $\\frac{1}{n} \\sum_v d(v)^2 \\ge \\left(\\frac{1}{n}\\sum_v d(v)\\right)^2$ for the degrees $d(v)$ of $G$, which holds by the Cauchy–Schwarz inequality.\n\n## Explanations\n\n### L1 human\n\nFix a small pattern $H$ with no odd cycle, such as a square.\nAmong all graphs with a given proportion of edges, which one has the fewest copies of $H$?\nSidorenko's conjecture says that a random graph is always at least as good as anything else: no clever arrangement of the edges can beat it.\n",
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      "posed_by": "Alexander Sidorenko"
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    "kind": "conjecture",
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    "name": "Sidorenko's conjecture",
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  "msc": [
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