Colouring set families without monochromatic k-chains

Abstract

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. *

Integer colorings with forbidden rainbow sums

“…4 < α − εn 1000 , and in either case we get a contradiction with (9) or (10). Thus, we have 9) and Lemma 4.3, we obtain that (X 3 + X 3 ) ∩ X 2 = ∅, and this contradicts (5).…”

“…and for every i ∈ J ′ 1 , there is at least one element in the pair {i, i + β} that is not contained in X 2 , we have that |J 1 ∩ X 2 | ≤ α − dn 2 = α − εn 800 , contradicts (9). Hence we may assume |J 2 ∩ X 3 | ≤ dn.…”

Integer colorings with no rainbow $k$-term arithmetic progression

“…The related problem of counting colorings of discrete objects with certain properties, was initiated by Erdős-Rothschild [10] who were interested in n-vertex graphs that admit the maximum number of redge-colorings without a given monochromatic subgraph, which has motivated a number of results (see, e.g., [1,2,16,23]). For the Erdős-Rothschild problems on other discrete structures, we refer to [20] for k-uniform hypergraphs, to [8] for intersecting set families, to [9] for k-chains and to [15,22] for sum-free sets. Meanwhile, the rainbow Erdős-Rothschild problem has received considerable attention, which studies the maximum number of r-edge-colorings of n-vertex graphs without a given rainbow subgraph whose edges are assigned distinct colors.…”