An Unstable Hypergraph Problem with a Unique Optimal Solution

Hoppen, Carlos; Kohayakawa, Yoshiharu; Lefmann, Hanno

doi:10.1007/978-3-642-36899-8_20

Cited by 2 publications

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“…For four colours, our method allows us to determine the optimal families precisely when t = 1. However, for t ≥ 2, we only get the asymptotics of the maximum number of (4, t)-colourings, while the precise characterisation obtained in[16] (see[17] for the full proof) requires careful study of the atypical colourings as well.…”

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confidence: 99%

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Colourings without monochromatic disjoint pairs

Clemens

Das

Tran

2018

European Journal of Combinatorics

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The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erdős-Rothschild problem, introduced in 1974 by Erdős and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure.The celebrated Erdős-Ko-Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erdős-Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces.In this paper we present a unified approach to the Erdős-Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erdős-Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann.1 To simplify the statements of our general results, we do not differentiate between sets, vector spaces or permutations in our notation. As Poincaré noted, "Mathematics is the art of giving the same name to different things."2 With a little more work, one can often show uniqueness.3 Their bounds seem to require n = Ωr,t k t 2 +t+1 .

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“…Given integers s ≥ 2, 1 ≤ ℓ ≤ s 2 , and 2 ≤ m j ≤ s, 1 ≤ j ≤ ℓ, satisfying the constraint ℓ we have ℓ j=1 (m j − 1 − log 3 m j ) ≥ s − 1 − log 3 s, with equality if and only if ℓ = 1 and m 1 = s. Now suppose we have at least two distinct (2t)-dimensional subspaces U j . By(17), we must have log 3…”

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confidence: 99%