TY  - JOUR
ID  - Cyman2014
LB  - Cyman2014
AU  - Cyman, Joanna
AU  - Dzido, Tomasz
TI  - A note on on-line Ramsey numbers for quadrilaterals
ST  - A note on on-line Ramsey numbers for quadrilaterals
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2014
VL  - 34
IS  - 3
SP  - 463
EP  - 468
DO  - http://dx.doi.org/10.7494/OpMath.2014.34.3.463
UR  - http://dx.doi.org/10.7494/OpMath.2014.34.3.463
AB  - We consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder's goal is to force Painter to create a monochromatic copy of a fixed graph \(H\) in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number \(\widetilde{r}(H)\) of the graph \(H\). An asymmetric version of the on-line Ramsey numbers \(\widetilde{r}(G,H)\) is defined accordingly. In 2005, Kurek and Ruciński computed \(\widetilde{r}(C_3)\). In this paper, we compute \(\widetilde{r}(C_4,C_k)\) for \(3 \le k \le 7\). Most of the results are based on computer algorithms but we obtain the exact value \(\widetilde{r}(C_4)\) and do so without the help of computer algorithms.
KW  - Ramsey theory
KW  - on-line games
SN  - 1232-9274
ER  -