Edge-colorings of uniform hypergraphs avoiding monochromatic matchings

Hoppen, Carlos; Kohayakawa, Yoshiharu; Lefmann, Hanno

doi:10.1016/j.disc.2014.10.004

Cited by 7 publications

(3 citation statements)

References 12 publications

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“…When t ≥ 2, one still has to specify which t-stars to take, as there are non-isomorphic choices, and Hoppen, Kohayakawa and Lefmann gave a precise description of the optimal families. Some stability results were also obtained, and in a later paper [19], the same authors consider the problem where one does not forbid monochromatic disjoint pairs, but rather matchings of larger size.…”

Section: Erdős-rothschild For Intersecting Familiesmentioning

confidence: 91%

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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Section: Erdős-rothschild For Intersecting Familiesmentioning

confidence: 91%

Colourings without monochromatic disjoint pairs

Clemens

Das

Tran

2018

European Journal of Combinatorics

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“…Lefmann, Person, Rödl and Schacht [19] studied the problem in the setting of three-uniform hypergraphs, with monochromatic copies of the Fano plane forbidden, before Lefmann, Person and Schacht [20] considered arbitrary k-uniform hypergraphs. Further results along this line of research can be found in [12,13,14,18]. Moving the problem into the domain of extremal set theory, Hoppen, Kohayakawa and Lefmann [15] solved the Erdős-Rothschild extension of the famous Erdős-Ko-Rado Theorem [9].…”

Section: Erdős-rothschild Problemsmentioning

confidence: 99%

Colouring set families without monochromatic k-chains

Das

Glebov

Sudakov

et al. 2019

Journal of Combinatorial Theory, Series A

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A coloured version of classic extremal problems dates back to Erdős and Rothschild, who in 1974 asked which n-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erdős-Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erdős' extension to k-chain-free families.Given a family F of subsets of [n], we define an (r, k)-colouring of F to be an r-colouring of the sets without any monochromatic k-chains F 1 ⊂ F 2 ⊂ . . . ⊂ F k . We prove that for n sufficiently large in terms of k, the largest k-chain-free families also maximise the number of (2, k)-colourings. We also show that the middle level, [n] ⌊n/2⌋ , maximises the number of (3, 2)-colourings, and give asymptotic results on the maximum possible number of (r, k)-colourings whenever r(k − 1) is divisible by three. *

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“…Determining the extremal configurations in general for k ≥ 2 and r ≥ 4 turned out to be a difficult problem. For further results along this line of research (when F is a non-complete graph or a hypergraph), we refer to [16,18,19,26,27,28].…”

Section: Erdős-rothschild Problems In Various Settingsmentioning

confidence: 99%

Integer colorings with forbidden rainbow sums

Cheng¹,

Jing²,

Li³

et al. 2020

Preprint

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For a set of positive integers A ⊆ [n], an r-coloring of A is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothchild problem in the context of sum-free sets, which asks for the subsets of [n] with the maximum number of rainbow sum-free r-colorings. We show that for r = 3, the interval [n] is optimal, while for r ≥ 8, the set [⌊n/2⌋, n] is optimal. We also prove a stability theorem for r ≥ 4. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.

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Edge-colorings of uniform hypergraphs avoiding monochromatic matchings

Cited by 7 publications

References 12 publications

Colourings without monochromatic disjoint pairs

Colourings without monochromatic disjoint pairs

Colouring set families without monochromatic k-chains

Integer colorings with forbidden rainbow sums

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