The number of monochromatic solutions to multiplicative equations

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• Victor Souza (Cambridge)
• Thursday 24 October 2024, 14:30-15:30
• MR12.

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Given an r-colouring of the interval {2,...,N}, what is the minimum number of monochromatic solutions of the equation xy = z? For r = 2, we show that there are always asymptotically at least (1/2sqrt(2)) N ^{log(N) monochromatic solutions, and that the leading constant is sharp. We also establish a stability version of this result. For general r, we show that there are at least C_r N}(1/S(r-1)) monochromatic solutions, where S® is the Schur number for r colours and C_r is a constant. This bound is sharp up to logarithmic factors when r <= 4. We also obtain results for more general multiplicative equations.

This talk is part of the Combinatorics Seminar series.

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