Current Research

(a) Perhaps the most important topic in extremal graph theory, is to determine the Turán number of a graph H . This is the maximal number of edges a graph of size n can have, without having a subgraph isomorphic to H . When H is non-bipartite, the asymptotics of this function is determined by the celebrated Erdös-Stone theorem, up to an 1 + o(1) factor. However, it is a longstanding problem to find good estimates on the Turán number of bipartite graphs, and in particular of even length cycles, C2k. In my past work in [1, 3, 4], I developed a technique for obtaining lower bounds on extremal properties of graphs, using entropy arguments. Recently I found one more application for this technique, in a joint work with Jacques A. Verstraete, and Felix Lazebnik, concerning a conjecture of Sidorenko 1991: For any l < k, the number of length 2k cycles is maximized in a C2l free graph, when the graph is a Turán graph for C2l. We have made significant progress in proving the conjecture for the cases l = k − 1, and l = k− 2. Our main tool is a generalization of my work [1]. That work gives a lower bound on the number of locally one to one embeddings of the length k path into a graph of a specified edge density. Our recent generalization, is to prove a similar result when embedding trees instead of paths.