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For any graph F and any integer r ≥ 2, the online vertex-Ramsey density of F and r, denoted m ∗ (F, r), is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [arXiv:1103.5849], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs. the binomial random graph Gn,p). For a large class of graphs F, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for m ∗ (F, r) are known. In this work we show that for the case where F = Pℓ is a long path, the picture is very different. It is not hard to see that m ∗ (Pℓ, r) = 1−1/k ∗ (Pℓ, r) for an appropriately defined integer k ∗ (Pℓ, r), and that the greedy strategy gives a lower bound of k ∗ (Pℓ, r) ≥ ℓ r. We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in ℓ, and we show that no superpolynomial improvement is possible

Year: 2013

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