The authors prove an improved recursive upper bound for hypergraph vertex online Ramsey numbers that yields lower-order improvements to hypergraph Ramsey number bounds.
Three Proofs of the Hypergraph Ramsey Theorem (An exposition)
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abstract
Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the construction of Conlon-Fox-Sudakov and hence obtain a slightly better bound.
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2026 1verdicts
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Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers
The authors prove an improved recursive upper bound for hypergraph vertex online Ramsey numbers that yields lower-order improvements to hypergraph Ramsey number bounds.