Colourings without monochromatic disjoint pairs

Abstract

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 .

Integer colorings with forbidden rainbow sums

“…In order to obtain the same conclusion in Lemma 4.2, we further require that the size of A is significantly smaller than n, since if A is close to [n], when we color all the elements in A by two colors, the number of colorings we obtained is also close to the extremal case. Note that if we use the same proof as in Lemma 4.2 for r = 4, equation (7) does not give us the conclusion we want. Hence the proof of Lemma 4.4 requires a more careful and complicated analysis of the structures of the containers.…”

On the Maximum Number of Integer Colourings with Forbidden Monochromatic Sums

“…The authors of [8] and [19] considered the problem of counting the number of colourings of families of r-sets such that every colour class is -intersecting. A related result in the context of vector spaces over a finite field GF (q) is proved in [23].…”

Colouring set families without monochromatic k-chains

“…If F ′ is still in A j for some i ≤ j ≤ k − 2 (in particular, if F falls under Case 3), F ′ must have fallen under Case 1, and thus contains more than ω sets in A j+1 , thereby ensuring that the branching down part of the operation can also be carried out. 4 However, when we remove sets in the branching operation, we could destroy the qualities (Q3) and (Q5) of sets that we have already considered, as the up-and down-degrees could decrease, and hence we must restore these. For 1 ≤ i ≤ k − 1, we return any sets in D i to the part A i .…”

“…, C r ) is bounded by F ∈F t(F ). Some straightforward optimisation (see, for instance, Lemma 3.1 in [4]) shows that this expression is maximised subject to the upper bound on the sum when each t(F ) is equal to 3, and so c(C 1 , . .…”

“…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]. Hoppen, Lefmann and Odermann [16] provided initial results for the vector space analogue of the Erdős-Ko-Rado Theorem, before Clemens, Das and Tran [4] presented a unified proof extending some of these results. In the context of additive combinatorics, the Erdős-Rothschild extension for sum-free sets has been pursued by Hàn and Jiménez [10] for abelian groups and Liu, Sharifzadeh and Staden [21] for subsets of the integers.…”