# Turán number for even cycles

For a graph $$G$$, define the Turán number $$t(n, G)$$ to be the largest integer $$m$$ such that there exists a graph on $$n$$ vertices with $$m$$ edges that does not contain $$G$$ as a subgraph. In other words, if $$H$$ has $$n$$ vertices and more than $$t(n, G)$$ edges, then $$H$$ must contain $$G$$ as a subgraph.

It is easy to see that for odd cycles, the Turán number $$t(n,C_{2k+1})= \lfloor n^2/4 \rfloor$$ for $$n > 2k+1$$, since no bipartite graph contains an odd cycle. However, the problem of determining the Turán numbers for even cycles is still open.

For the $$4$$-cycle $$C_4$$, it was known [1] for some time that the Turán number $$t(n, C_4)$$ is of order $$n^{3/2}$$. In 1983, Füredi [2] determined the exact values of $$t(n, C_4)$$ for infinitely many $$n$$ (in particular, for $$n=2^k$$ for some $$k$$). In 1996, Füredi generalized this result [3]. He proved that for $$q \geq 15$$ and $$n=q^2+q+1$$,

$$\displaystyle t(n,C_4) \leq \frac{1}{2} q(q+1)^2.$$

In particular, for a prime power $$p \geq 13$$ and $$n=p^2+p+1$$,

$$\displaystyle t(n,C_4) = \frac{1}{2} p(p+1)^2.$$

Exact values of $$t(n, C_4)$$ are known for all $$n\leq 21$$ (see, for example, [4]).

For the general case, Erdös [5] and Bondy and Simonovits [6] showed that

$$\displaystyle t(n,C_{2k}) \leq c k n^{1+1/k}.$$

Erdös also posed the following conjecture:

# Conjecture (Proposed by Erdös[5])

Prove that $t(n,C_{2k}) \geq cn^{1+1/k}$ for $$k =4$$ and $$k \geq 6$$.

We note that this conjecture, together with the above result of Erdös, Bondy and Simonovits, would imply that $$t(n, C_{2k})=o_n(n^{1+1/k})$$. In fact, it has been conjectured by Erdös and Simonovits[7] that $$t(n, C_{2k})=(\frac{1}{2}+o(1))n^{1+1/k})$$. For the case that $$k=5$$, Lazebnik, Ustimenko, and Woldar [8] disprove this second conjecture, showing that $$t(n, C_{2k})>(c+o(1))n^{1+1/k})$$, where $$c\approx 0.58$$ (see also [9] for a further discussion on this counterexample). This was also disproven in the case that $$k=3$$ by Füredi, Naor, and Verstraëte [10].

A lower bound of order $$n^{1+1/(2k-1)}$$ for the above conjecture can be proved by probabilistic deletion methods [11]. The bipartite Ramanujan graph[12], [13] shows that $$t(n,C_{2k}) \geq n^{1+2/3k}$$. In 1995, Lazebnik, Ustimenko and Woldar [14] constructed graphs which yield $$t(n,C_{2k}) \geq n^{1+ 2 /(3k-3)}$$. The same authors [8] prove (by construction) that $$t(n, C_{2k})\geq (\frac{k-1}{k^{1+1/k}}+o(1))n^{1+1/k}$$ (it is this construction that yields the counterexample to Erdös and Simonovits' conjecture, as noted above).

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