Discussiones Mathematicae Graph Theory 25(1-2) (2005)
141-149
doi: 10.7151/dmgt.1268
Andrzej Kurek and Andrzej Ruciński
Department of Discrete Mathematics
Adam Mickiewicz University
Poznań, Poland
e-mail: kurek@amu.edu.pl
e-mail: rucinski@amu.edu.pl
In the first part of this paper we show that when H is a complete graph K_{k} on k vertices, then m_{inf}(H,r) = (R−1)/2, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál that the size Ramsey number for K_{k} equals (^{R}_{2}).
We also study an on-line version of the size Ramsey number, related to the following two-person game: Painter colors on-line the edges provided by Builder, and Painter's goal is to avoid a monochromatic copy of K_{k}. The on-line Ramsey number R(k;r) is the smallest number of moves (edges) in which Builder can force Painter to lose if r colors are available. We show that R(3;2) = 8 and R(k;2) ≤ 2k(^{2k}_{k−}^{−2}_{1}), but leave unanswered the question if R(k;2) = o(R^{2}(k;2)).
Keywords: size Ramsey number, graph density, online Ramsey games.
2000 Mathematics Subject Classification: 05C55, 05D10, 91A43.
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Received 16 November 2003