An Approximate Version of Sidorenko's Conjecture

作者：Conlon David^*; Fox Jacob; Sudakov Benny

来源：Geometric and Functional Analysis, 2010, 20(6): 1354-1366.

DOI：10.1007/s00039-010-0097-0

• Sidorenko's conjecture
• graph homomorphism
• subgraph density

A beautiful conjecture of ErdAs-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.