A lower bound for set-colouring Ramsey numbers

Hypergraph Ramsey numbers of cliques versus stars

“…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 r(K (3) 4 , S (3) n ) exhibits an unusual intermediate growth rate, namely,…”

“…Seminal work of Erdős-Rado [11] and Erdős-Hajnal (see, e.g., [6]) reduces the estimation of diagonal Ramsey numbers for k ≥ 3 to the k = 3 case. For off-diagonal Ramsey numbers, the only case for which the tower height of the growth rate is not known is r(K (k) k+1 , K (k) n ), though it was noted in [16] that this tower height could be determined by proving that the 4-uniform Ramsey number r(K (4) 5 , K (4) n ) is double exponential in a power of n. Moreover, it was shown in [15] that if r(K (3) n , K (3) n ) grows double exponentially in a power of n, then the same is also true for r(K (4) 5 , K (4) n ). Hence, the growth rate for all diagonal and off-diagonal hypergraph Ramsey numbers with k ≥ 4 would follow from knowing the growth rate of the diagonal Ramsey number when k = 3.…”

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Let K (3) m denote the complete 3-uniform hypergraph on m vertices and S (3) n the 3-uniform hypergraph on n + 1 vertices consisting of alledges 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 r(K (3) 4 , S (3) n ) exhibits an unusual intermediate growth rate, namely,for some positive constants c and c ′ .

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Let K (3) m denote the complete 3-uniform hypergraph on m vertices and S (3) n the 3-uniform hypergraph on n + 1 vertices consisting of alledges 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 r(K (3) 4 , S (3) n ) exhibits an unusual intermediate growth rate, namely,for some positive constants c and c ′ . The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum N such that any 2-edge-coloring of the Cartesian product K N □K N contains either a red rectangle or a blue K n ?

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“…Let K (3) m denote the complete 3-uniform hypergraph on m vertices and S (3) n the 3-uniform hypergraph on n + 1 vertices consisting of all…”