1,621 research outputs found
An approximate version of Sidorenko's conjecture
A beautiful conjecture of Erd\H{o}s-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.Comment: 12 page
Ramsey Goodness and Beyond
In a seminal paper from 1983, Burr and Erdos started the systematic study of
Ramsey numbers of cliques vs. large sparse graphs, raising a number of
problems. In this paper we develop a new approach to such Ramsey problems using
a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type
stability, and other structural results. We give exact Ramsey numbers for
various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
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