Botler, F., Corsten, J., Dankovics, A., Frankl, N., Hàn, H., Jiménez, A. and Skokan, J.
(2019)
*Maximum number of triangle-free edge colourings with five and six colours.*
Acta Mathematica Universitatis Comenianae, 88 (3).
pp. 495-499.
ISSN 0862-9544

Text (Maximum number of triangle-free colourings with five and six colours)
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## Abstract

Let k ≥ 3 and r ≥ 2 be natural numbers. For a graph G, let F(G, k, r) denote the number of colourings of the edges of G with colours 1,…, r such that, for every colour c ∈ {1,…, r}, the edges of colour c contain no complete graph on k vertices Kk. Let F(n, k, r) denote the maximum of F(G, k, r) over all graphs G on n vertices. The problem of determining F(n, k, r) was first proposed by Erdős and Rothschild in 1974, and has so far been solved only for r = 2; 3, and a small number of other cases. In this paper we consider the question for the cases k = 3 and r = 5 or r = 6. We almost exactly determine the value F(n, 3, 6) and approximately determine the value F(n, 3, 5) for large values of n. We also characterise all extremal graphs for r = 6 and prove a stability result for r = 5.

Item Type: | Article |
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Additional Information: | © 2019 Acta Mathematica Universitatis Comenianae |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 08 Nov 2019 15:45 |

Last Modified: | 28 May 2020 23:13 |

URI: | http://eprints.lse.ac.uk/id/eprint/102464 |

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