BY Dr. Artem Chernikov, Ph.D.
University of California, Los Angeles
Fix an arbitrary bipartite finite graph H, and let's consider the family of all finite graphs not containing H as an induced subgraph. Recently a strong form of Szemeredi's regularity lemma for such families was established by Lovasz and Szegedy. In particular, up to an arbitrary small error, such graphs can be uniformly approximated by direct products of unary relations. Using a combination of analytic, combinatorial and model-theoretic methods, we establish a generalization of this result for hypergraphs, showing that k-ary hypergraphs omitting a fixed k-partite hypergraph H can be uniformly approximated by relations of arity smaller than k. No prior knowledge of the aforementioned notions will be assumed. Joint work with Henry Towsner.
Professor Chernikov is an Associate Professor at UCLA. He received his doctorate in 2012 from the Université Claude Bernard - Lyon 1 and subsequently held postdoctoral positions at the Einstein Institute of Mathematics, Hebrew University of Jerusalem; the Mathematical Sciences Research Institute in Berkeley; and the Université Paris Diderot - Paris 7. His Ph.D. thesis received the Karp prize for the best thesis in logic internationally. His research has been supported by a Sloan Research Fellowship and an NSF Career Grant. It is published in top journals like the Journal of the American Mathematical Society and the Journal of the European Mathematical Society.