On the number of $k$-gons in finite projective planes

Let $\Pi$ be a projective plane of order $n$ and $\Gamma_{\Pi}$ be its Levi graph (the point-line incidence graph). For fixed $k \geq 3$, let $c_{2k}(\Gamma_{\Pi})$ denote the number of $2k$-cycles in $\Gamma_{\Pi}$. In this paper we show that $$c_{2k}(\Gamma_{\Pi}) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n \rightarrow \infty.$$ We also state a conjecture regarding the third and fourth largest terms in the asymptotic of the number of $2k$-cycles in $\Gamma_{\Pi}$. This result was also obtained independently by Voropaev in 2012. Let $\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\})$ denote the greatest number of $2k$-cycles amongst all bipartite graphs of order $v$ and girth at least 6. As a corollary of the result above, we obtain $$\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\}) = \left(\frac{1}{2^{k+1}k}-o(1)\right)v^k, \hspace{0.5cm} v \rightarrow \infty.$$