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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 ?

 

We prove that the number of edges of a multigraph with vertices is at most , provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi, and Leighton, if has edges, in any drawing of with the above property, the number of crossings is . This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.

 

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. 

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least (logn)1/4−o(1) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound Ω(log1/8n) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least (logn)1−o(1) . 

We show that there is an absolute constant c > 0 c>0 such that the following holds. For every n > 1 n > 1 , there is a 5-uniform hypergraph on at least 2 2 c n 1 / 4 2^{2^{cn^{1/4}}} vertices with independence number at most n n , where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for k k -uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.