Casey Tompkins, On graphs without cycles of length 0 modulo 4

Bollobás proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$ contains an even number, an $n$-vertex graph containing no cycle of length $\ell \bmod k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs $\ell$ and $k$. We precisely determine the maximum number of edges in a graph containing no cycle of length $0 \bmod 4$.
This is joint work with Ervin Győri, Binlong Li, Nika Salia, Kitti Varga and Manran Zhu.