Proves the exact Ramsey number R^arith_2 equals 9 for monochromatic triples in E_n of B_n and establishes 2^{δn+o(n)} ≤ M^arith_2(B_n) ≤ 2^{γn+o(n)} with explicit entropy constants δ≈1.356779 and γ≈1.567837.
Off-Diagonal Ramsey Multiplicity
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abstract
The Ramsey multiplicity problem asks for the minimum asymptotic density of monochromatic labelled copies of a graph $H$ in a red/blue colouring of the edges of $K_n$. We introduce an off-diagonal generalization in which the goal is to minimize a certain weighted sum of the densities of red copies of one graph and blue copies of another. We build up various properties of this new notion, including a useful "dual formulation," and use these results to solve the problem for several pairs of graphs.
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math.CO 1years
2026 1verdicts
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Multiplicity for partially ordered sets
Proves the exact Ramsey number R^arith_2 equals 9 for monochromatic triples in E_n of B_n and establishes 2^{δn+o(n)} ≤ M^arith_2(B_n) ≤ 2^{γn+o(n)} with explicit entropy constants δ≈1.356779 and γ≈1.567837.