# Size Ramsey number for complete balanced bipartite graphs

The size Ramsey number $$\hat{r}(G,H)$$ is the least integer $$m$$ for which there exists a graph $$F$$ with $$m$$ edges so that in any coloring of the edges of $$F$$ in red and blue, there is always either a red copy of $$G$$ or a blue copy of $$H$$. Sometimes we write $$F \rightarrow (G,H)$$ to denote this. For $$G=H$$, we denote $$\hat{r}(G,G)$$ by $$\hat{r}(G)$$.

For the complete graph $$K_n$$, Erdös, Faudree, Rousseau and Schelp [1] proved that

$$\displaystyle \hat{r}(K_n) = \binom {r(n)}{2}.$$

They asked the following size Ramsey problem for $$K_{n,n}$$:

# Problem

Determine $$\hat{r}(K_{n,n})$$.

Erdös, Faudree, Rousseau and Schelp [4] and Nešetril and Rödl [3] proved the following upper bound for $$\hat{r}(K_{n,n})$$:

$$\displaystyle \hat{r}(K_{n,n}) < \frac 3 2 n^3 2^n.$$

For the lower bound, Erdös and Rousseau [2] proved by probabilistic methods that for $$n \geq 6$$,

$$\displaystyle \hat{r}(K_{n,n}) > \frac 1 {60} n^2 2^n.$$

Bibliography
1 P. Erdös, R. Faudree, C. C. Rousseau and R. H. Schelp, Ramsey numbers for brooms, Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982), Congr. Numer. 35 (1982), 283-293.

2 P. Erdös and C. C. Rousseau. The size Ramsey number of a complete bipartite graph, Discrete Math. 113 (1993), 259-262.

3 J. Nešetril and V. Rödl. The structure of critical graphs, Acta. Math. cad. Sci. Hungar. 32 (1978), 295-300.

4 P. Erdös, R. J. Faudree, C. C. Rousseau and R. H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978), 145-161.