combinatorics / Arithmetic Ramsey Theory
What is the minimum asymptotic density $\delta_k$ of monochromatic $k$-term arithmetic progressions in every two-colouring of $\{1, \dots, n\}$? The exact certificate gives $\delta_3 = 117/2192$, matching the known 548-bead colouring.
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exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open
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What is the minimum asymptotic density $\delta_k$ of monochromatic $k$-term arithmetic progressions in every two-colouring of $\{1, \dots, n\}$? The exact certificate gives $\delta_3 = 117/2192$, matching the known 548-bead colouring.
exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open
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