# Turán number for bipartite graphs is a rational power of n

Let $$G$$ be a graph. We define the Turán number $$t(n, G)$$ to be the minimum integer $$m$$ so that there exists a graph on $$n$$ vertices with $$m$$ edges without $$G$$ as a subgraph.

For a general graph $$G$$ with chromatic number $$\chi(G)\geq 3$$, the Erdos-Simonovits-Stone Theorem [1] can be used to determine $$t(n, G)$$ asymptotically:

Theorem 1 (The Erdos-Simonovits-Stone Theorem)   For a graph $$G$$ with chromatic number $$\chi(G)\geq 3$$, the Turán number $$t(n, G)$$ satisfies

$$\displaystyle t(n,G) \geq \left(1- \frac{1}{\chi(G) -1}\right) {n\choose 2} + o(n^2)$$

This result was further strengthened by Bollobás, Erdos, and Simonovits[2], and Chvátal and Szemerédi [3] as follows:

Theorem 2   For any $$\epsilon >0$$, a graph with

$$\displaystyle \left(1-\frac{1}{p}+\epsilon\right) {n\choose 2}$$

edges must contain a complete $$(p+1)$$-partite graph with each part consisting of $$m$$ vertices where

$$\displaystyle m > c \frac{\log n}{ \log 1/\epsilon}.$$

The only case that has eluded the power of the above theorems is the case where $$\chi(G)=2$$, that is when $$G$$ is a bipartite graph. The following problems were proposed in [4].

# Conjecture

For all rationals $$1<\frac{p}{q}<2$$, there exists a bipartite graph $$G$$ such that $$t(n, G)=\Theta(n^{p/q})$$.

# Problem

Given a bipartite graph $$G$$, does there exist a rational exponent $$r=r(G)$$ such that $$t(n, G)=\Theta(n^r)$$?

Bibliography
1 P. Erdos and M. Simonovits. A limit theorem in graph theory. Studia Sci. Math. Hungar. 1 (1966), 51-57.

2 B. Bollobás, P. Erdos and M. Simonovits. On the structure of edge graphs, II. J. London Math. Soc. (2) 12 (1975/76) no.2, 219-224.

3 V. Chvátal and E. Szemerédi. On the Erdos-Stone theorem. Journal of London Math. Soc. Ser. 2, 23 (1981), 207-214.

4 P. Erdös and M. Simonovits, Cube-supersaturated graphs and related problems, Progress in graph theory (Waterloo, Ont., 1982), 203-218, Academic Press, Toronto, 1984.