Seminars in January 2013

  • Po-Shen Loh, The critical window for the Ramsey-Turan problem

    The critical window for the Ramsey-Turan problem
    Po-Shen Loh
    Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA
    2013/1/11 Fri 4PM-5PM
    The first application of Szemeredi’s regularity method was the following celebrated Ramsey-Turan result proved by Szemeredi in 1972: any K4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N2 edges. Four years later, Bollobas and Erdos gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K4-free graph on N vertices with independence number o(N) and (1/8 – o(1)) N2 edges. Starting with Bollobas and Erdos in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this work, we give nearly best-possible bounds, solving the various open problems concerning this critical window.
    More generally, it remains an important problem to determine if, for certain applications of the regularity method, alternative proofs exist which avoid using the regularity lemma and give better quantitative estimates. In their survey on the regularity method, Komlos, Shokoufandeh, Simonovits, and Szemeredi surmised that the regularity method is likely unavoidable for applications where the extremal graph has densities in the regular partition bounded away from 0 and 1. In particular, they thought this should be the case for Szemeredi’s result on the Ramsey-Turan problem. Contrary to this philosophy, we develop new regularity-free methods which give a linear dependence, which is tight, between the parameters in Szemeredi’s result on the Ramsey-Turan problem.
    Joint work with Jacob Fox and Yufei Zhao.
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  • Choongbum Lee (이중범), Ramsey numbers of cubes versus cliques

    Ramsey numbers of cubes versus cliques
    Choongbum Lee (이중범)
    Department of Mathematics, MIT, Cambridge, MA, USA
    2013/01/04 Fri 4PM-5PM
    The cube graph Qn is the skeleton of the n-dimensional cube. It is an n-regular graph on 2n vertices. The Ramsey number r(Qn, Ks) is the minimum N such that every graph of order N contains the cube graph Qn or an independent set of order s. Burr and Erdős in 1983 asked whether the simple lower bound r(Qn, Ks) ≥ (s-1)(2n -1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.
    Joint work w/ David Conlon, Jacob Fox, and Benny Sudakov
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