The Erd\H{o}s-Gallai bound for consecutive even cycle lengths
Erd\H{o}s and Gallai in 1959 proved the seminal result that every $n$-vertex graph with no cycle of length at least $2t+2$ has at most $\tfrac{2t+1}{2}(n-1)$ edges. We prove the extension that, for every sufficiently large $t$, the same quantity is also the sharp extremal bound for graphs with no $t$ consecutive even cycle lengths, resolving a conjecture of Verstra\"ete. Thus, at the Erd\H{o}s-Gallai threshold, forcing an entire interval of even cycle lengths costs no more than forcing its longest member. More precisely, every $n$-vertex graph $G$ with $e(G)\ge \tfrac{(2t+1)(n-1)}2$ $\bullet$ either contains $t$ consecutive even cycle lengths, $\bullet$ or equality holds and $G$ is connected with every block isomorphic to $K_{2t+1}$. As consequences, for every sufficiently large even $k$ we determine the sharp edge thresholds forcing a cycle of length $0\pmod k$ or $2\pmod k$, answering questions of Bai, Grzesik, Li, and Prorok and of Gao, Li, Ma and Xie, respectively, for sufficiently large even $k$. The proof develops a stability-enhanced sublinear expander method. Its main new ingredient is a dense-case decomposition that recovers the lengths lost in the expander extraction by combining a flexible dense core with rooted cycle families in the vertices outside the core.