On-line Chain Partitions of Up-growing Semi-orders
Stefan Felsner, Kamil Kloch, Grzegorz Matecki, Piotr Micek · Order · 2011
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 val(w) such that Algorithm has a strategy using at most val(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 $\big\lfloor\frac{1+\sqrt{5}}{2}\; w \big\rfloor$ .