Abstract
A beautiful conjecture of Erdős-Simonovits and Sidorenko states that, if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.
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D.C.’s research supported by a Junior Research Fellowship at St John’s College. J.F.’s research supported by an NSF Graduate Research Fellowship and a Princeton Centennial Fellowship. B.S.’s research supported in part by NSF CAREER award DMS-0812005 and by a USA-Israeli BSF grant.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Conlon, D., Fox, J. & Sudakov, B. An Approximate Version of Sidorenko’s Conjecture. Geom. Funct. Anal. 20, 1354–1366 (2010). https://doi.org/10.1007/s00039-010-0097-0
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DOI: https://doi.org/10.1007/s00039-010-0097-0