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- W2963549612 abstract "Let R ( k 1 , … , k r ) denote the classical r -color Ramsey number for integers k i ≥ 2 . The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if k 1 , … , k r are integers no smaller than 3 and k r − 1 ≤ k r , then R ( k 1 , … , k r − 2 , k r − 1 − 1 , k r + 1 ) ≤ R ( k 1 , … , k r ) . We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let R r ( k ) stand for the r -color Ramsey number R ( k , … , k ) . It is known that lim r → ∞ R r ( 3 ) 1 ∕ r exists, either finite or infinite, the latter conjectured by Erdős. This limit is related to the Shannon capacity of complements of K 3 -free graphs. We prove that if DC holds, and lim r → ∞ R r ( 3 ) 1 ∕ r is finite, then lim r → ∞ R r ( k ) 1 ∕ r is finite for every integer k ≥ 3 ." @default.
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- W2963549612 date "2019-08-01" @default.
- W2963549612 modified "2023-09-27" @default.
- W2963549612 title "On a Diagonal Conjecture for classical Ramsey numbers" @default.
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- W2963549612 doi "https://doi.org/10.1016/j.dam.2019.07.006" @default.
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