Monochromatic Progressions in Random Colorings

Let N^{+}(k)= 2^{k/2} k^{3/2} f(k) and N^{-}(k)= 2^{k/2} k^{1/2} g(k) where 1=o(f(k)) and g(k)=o(1). We show that the probability of a random 2-coloring of {1,2,…,N^{+}(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1,2,…,N^{-}(k)} containing a monochromatic k-term arithmetic progression approaches 0, for large k. This improves an upper bound due to Brown, who had established an analogous result for N^{+}(k)= 2^k log k f(k).

💡 Research Summary

The paper investigates the probabilistic behavior of monochromatic arithmetic progressions (APs) in random two‑colorings of the integer interval ({1,2,\dots ,N}). Classical Ramsey theory guarantees that for any fixed length (k) there exists a deterministic Van der Waerden number (W(k)) such that every 2‑coloring of ({1,\dots ,W(k)}) contains a monochromatic (k)-term AP. However, the exact growth of (W(k)) is unknown, and the paper asks a probabilistic analogue: how large must (N) be for a random 2‑coloring to contain a monochromatic (k)-AP with probability tending to 1, and how small can (N) be while the probability still tends to 0 as (k\to\infty)?

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