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For positive integers 𝑛, 𝑟, 𝑠 with 𝑟 > 𝑠, the set-coloring Ramsey number 𝑅(𝑛; 𝑟, 𝑠) is the minimum 𝑁 such that if every edge of the complete graph 𝐾_𝑁 receives a set of 𝑠 colors from a palette of 𝑟 colors, then there is guaranteed to be a monochromatic clique on 𝑛 vertices, that is, a subset of 𝑛 vertices where all of the edges between them receive a common color. In particular, the case 𝑠 = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on 𝑅(𝑛; 𝑟, 𝑠) which imply that 𝑅(𝑛; 𝑟, 𝑠) = 2^Θ(𝑛𝑟) if 𝑠/𝑟 is bounded away from 0 and 1. The upper bound extends an old result of Erdős and Szemerédi, who treated the case 𝑠 = 𝑟 − 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.more » « less
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Mulzer, Wolfgang; Phillips, Jeff M (Ed.)We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges.more » « less
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Abstract Let denote the complete 3‐uniform hypergraph on vertices and the 3‐uniform hypergraph on vertices consisting of all edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off‐diagonal Ramsey number exhibits an unusual intermediate growth rate, namely,for some positive constants and . The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum such that any 2‐edge‐coloring of the Cartesian product contains either a red rectangle or a blue ?more » « less
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Abstract The book graph $$B_n ^{(k)}$$ consists of $$n$$ copies of $$K_{k+1}$$ joined along a common $$K_k$$ . In the prequel to this paper, we studied the diagonal Ramsey number $$r(B_n ^{(k)}, B_n ^{(k)})$$ . Here we consider the natural off-diagonal variant $$r(B_{cn} ^{(k)}, B_n^{(k)})$$ for fixed $$c \in (0,1]$$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $$c$$ , a simple $$k$$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $$k$$ -partite, while, for $$c$$ bounded away from $$0$$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $$c$$ .more » « less
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