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- W4288289968 abstract "We prove Tur'an-type theorems for two related Ramsey problems raised by Bollob'as and by Fox and Sudakov. First, for $t ge 3$, we show that any two-colouring of the complete graph on $n$ vertices that is $delta$-far from being monochromatic contains an emph{unavoidable $t$-colouring} when $delta gg n^{-1/t}$, where an unavoidable $t$-colouring is any two-colouring of a clique of order $2t$ in which one colour forms either a clique of order $t$ or two disjoint cliques of order $t$. Next, for $ tge 3$, we show that any tournament on $n$ vertices that is $delta$-far from being transitive contains an emph{unavoidable $t$-tournament} when $delta gg n^{-1/lceil t/2 rceil}$, where an unavoidable $t$-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order $t$. Conditional on a well-known conjecture about bipartite Tur'an numbers, both results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers." @default.
- W4288289968 created "2022-07-28" @default.
- W4288289968 creator A5045393655 @default.
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- W4288289968 date "2019-07-01" @default.
- W4288289968 modified "2023-10-18" @default.
- W4288289968 title "Tur'an theorems for unavoidable patterns" @default.
- W4288289968 doi "https://doi.org/10.48550/arxiv.1907.00964" @default.
- W4288289968 hasPublicationYear "2019" @default.
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