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Discussiones Mathematicae Graph Theory 25(1-2) (2005) 141-149
DOI: https://doi.org/10.7151/dmgt.1268

TWO VARIANTS OF THE SIZE RAMSEY NUMBER

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

Abstract

Given a graph H and an integer r ≥ 2, let G→ (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = max_{F ⊆ G}|E(F)|/|V (F)| and define the Ramsey density m_inf(H,r) as the infimum of m(G) over all graphs G such that G→ (H,r).

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 (^R₂).

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−⁻²₁), but leave unanswered the question if R(k;2) = o(R²(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