Even circuit theorem

In extremal graph theory, the even circuit theorem is a result of Paul Erdős according to which an $n$-vertex graph that does not have a simple cycle of length $2k$ can only have $O(n^{1 + 1/k})$ edges. For instance, 4-cycle-free graphs have $O(n^{3/2})$ edges, 6-cycle-free graphs have $O(n^{4/3})$ edges, etc.

History
The result was stated without proof by Erdős in 1964. published the first proof, and strengthened the theorem to show that, for $n$-vertex graphs with $&Omega;(n^{1 + 1/k})$ edges, all even cycle lengths between $2k$ and $2kn^{1/k}$ occur.

Lower bounds
The bound of Erdős's theorem is tight up to constant factors for some small values of k: for k = 2, 3, or 5, there exist graphs with $&Omega;(n^{1 + 1/k})$ edges that have no $2k$-cycle.

It is unknown for $k$ other than 2, 3, or 5 whether there exist graphs that have no $2k$-cycle but have $&Omega;(n^{1 + 1/k})$ edges, matching Erdős's upper bound. Only a weaker bound is known, according to which the number of edges can be $&Omega;(n^{1 + 2/(3k &minus; 3)})$ for odd values of $k$, or $&Omega;(n^{1 + 2/(3k &minus; 4)})$ for even values of $k$.

Constant factors
Because a 4-cycle is a complete bipartite graph, the maximum number of edges in a 4-cycle-free graph can be seen as a special case of the Zarankiewicz problem on forbidden complete bipartite graphs, and the even circuit theorem for this case can be seen as a special case of the Kővári–Sós–Turán theorem. More precisely, in this case it is known that the maximum number of edges in a 4-cycle-free graph is
 * $$n^{3/2}\left(\frac{1}{2}+o(1)\right).$$

conjectured that, more generally, the maximum number of edges in a $2k$-cycle-free graph is However, later researchers found that there exist 6-cycle-free graphs and 10-cycle-free graphs with a number of edges that is larger by a constant factor than this conjectured bound, disproving the conjecture. More precisely, the maximum number of edges in a 6-cycle-free graph lies between the bounds
 * $$n^{1+1/k}\left(\frac{1}{2}+o(1)\right).$$
 * $$0.5338n^{4/3} \le \operatorname{ex}(n,C_6) \le 0.6272n^{4/3},$$

where $ex(n,G)$ denotes the maximum number of edges in an $n$-vertex graph that has no subgraph isomorphic to $G$. The maximum number of edges in a 10-cycle-free graph can be at least
 * $$4\left(\frac{n}{5}\right)^{6/5} \approx 0.5798 n^{6/5}.$$

The best proven upper bound on the number of edges, for $2k$-cycle-free graphs for arbitrary values of $k$, is
 * $$n^{1+1/k}\left(k-1+o(1)\right).$$