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Published December 2020 | Submitted
Journal Article Open

Short proofs of some extremal results III


We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.

Additional Information

© 2020 Wiley Periodicals LLC. Issue Online: 25 October 2020; Version of Record online: 30 August 2020; Manuscript accepted: 02 June 2020; Manuscript received: 20 October 2019. This research was supported by the ERC Starting Grant, 676632 (D.C.). NSF Award, DMS‐1855635. Alfred P. Sloan Fellowship (J.F.), SNSF grant, 200021‐175573 (B.S.). Section 2 was first written in May 2015, predating a recent paper of Geneson [41] showing that T_k ≤ k^(5/2+o(1)). More recently J. Balogh, W. Linz and L. Mattos [9] independently investigated the question of estimating T_k and showed that T_k = k^(2+o(1)) (which is slightly weaker than Corollary 2.4). We would like to thank Kevin Ford for some helpful discussions on this theme and also thank the anonymous referees for their careful reading of the paper and useful suggestions.

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