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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Turán number of bipartite graphs with no $K_{t,t}$
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by Benny Sudakov and István Tomon PDF
Proc. Amer. Math. Soc. 148 (2020), 2811-2818 Request permission

Abstract:

The extremal number of a graph $H$, denoted by $\textrm {ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices that does not contain $H$. The celebrated Kővári-Sós-Turán theorem says that for a complete bipartite graph with parts of size $t\leq s$ the extremal number is $\textrm {ex}(K_{s,t})=O(n^{2-1/t})$. It is also known that this bound is sharp if $s>(t-1)!$. In this paper, we prove that if $H$ is a bipartite graph such that all vertices in one of its parts have degree at most $t$ but $H$ contains no copy of $K_{t,t}$, then $\textrm {ex}(n,H)=o(n^{2-1/t})$. This verifies a conjecture of Conlon, Janzer, and Lee.
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Additional Information
  • Benny Sudakov
  • Affiliation: Department of Mathematics, ETH Zurich, Rämistrasse 101, HG G 33.4, 8092 Zurich, Switzerland
  • MR Author ID: 602546
  • Email: benjamin.sudakov@math.ethz.ch
  • István Tomon
  • Affiliation: Department of Mathematics, ETH Zurich, Rämistrasse 101, HG G 33.4, 8092 Zurich, Switzerland
  • Email: istvan.tomon@math.ethz.ch
  • Received by editor(s): October 24, 2019
  • Published electronically: April 9, 2020
  • Additional Notes: Research supported by SNSF grant 200021-175573.
  • Communicated by: Patricia Hersh
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 2811-2818
  • MSC (2010): Primary 05C35
  • DOI: https://doi.org/10.1090/proc/15042
  • MathSciNet review: 4099770