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arXiv · 0704.1829

On-line Chain Partitions of Up-growing Semi-orders

Stefan Felsner·Kamil Kloch·Grzegorz Matecki·Piotr Micek
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Abstract

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents a partially ordered set, point by point. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width $w$ is a minimum number $\fVal(w)$ such that Algorithm has a strategy using at most $\fVal(w)$ chains on orders of width at most $w$. We analyze the chain partition game for up-growing semi-orders. Surprisingly, the golden ratio comes into play and the value of the game is $\lfloor\frac{1+\sqrt{5}}{2}\; w \rfloor$.

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Stefan Felsner, Kamil Kloch, Grzegorz Matecki, Piotr Micek. 2011-02-19. On-line Chain Partitions of Up-growing Semi-orders. https://arxiv.org/abs/0704.1829

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