The extremal function for cycles of length $\mathrm{\ell }$ mod $k$

Burr and Erd\H{o}s conjectured that for each $k,\mathrm{\ell }\in {\mathbb{Z}}^{+}$ such that $k\mathbb{Z}+\mathrm{\ell }$ contains even integers, there exists $c_k(\mathrm{\ell })$ such that any graph of average degree at least $c_k(\mathrm{\ell })$ contains a cycle of length $\mathrm{\ell }$ mod $k$. This conjecture was proved by Bollob'{a}s, and many successive improvements of upper bounds on $c_k(\mathrm{\ell })$ appear in the literature. In this short note, for $1\le \mathrm{\ell }\le k$, we show that $c_k(\mathrm{\ell })$ is proportional to the largest average degree of a $C_{\mathrm{\ell }}$-free graph on $k$ vertices, which determines $c_k(\mathrm{\ell })$ up to an absolute constant. In particular, using known results on Tur'{a}n numbers for even cycles, we obtain $c_k(\mathrm{\ell })=O(\mathrm{\ell }{k}^{2\mathrm{/}\mathrm{\ell }})$ for all even $\mathrm{\ell }$, which is tight for $\mathrm{\ell }\in \{4,6,10\}$. Since the complete bipartite graph $K_{\ell - 1,n - \ell

• 1}$hasnocycleoflength$2\ell$mod$k$,italsoshows$c_k(\ell) = \Theta(\ell)$for$\ell = \Omega(\log k)$.