Girao, Antonio;
Illingworth, Freddie;
Michel, Lukas;
Savery, Michael;
Scott, Alex;
(2024)
Flashes and Rainbows in Tournaments.
Combinatorica
, 44
pp. 675-690.
10.1007/s00493-024-00090-7.
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Abstract
Colour the edges of the complete graph with vertex set {1,2,⋯,n} with an arbitrary number of colours. What is the smallest integer f(l, k) such that if n>f(l,k) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that f(l,k)=lk-1 and proved this for l≥(3k)2k. We prove the conjecture for l≥k3(logk)1+o(1) and establish the general upper bound f(l,k)≤k(logk)1+o(1)·lk-1. This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.
| Type: | Article |
|---|---|
| Title: | Flashes and Rainbows in Tournaments |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00493-024-00090-7 |
| Publisher version: | http://dx.doi.org/10.1007/s00493-024-00090-7 |
| Language: | English |
| Additional information: | This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| Keywords: | Science & Technology, Physical Sciences, Mathematics, Ramsey theory, Paths, Tournaments |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10197006 |
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