Linear cycles of consecutive lengths

A well-known result of Verstra\"ete \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of $G$. We establish two extensions of Verstra\"ete's result for linear cycles in linear $r$-uniform hypergraphs. We show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_1=c_1(r)$ and $c_2=c_2(r,k)$, such that every linear $r$-uniform hypergraph $G$ with average degree $d(G)\geq c_1 k$ contains linear cycles of $k$ consecutive even lengths, the shortest of which is at most $2\lceil \frac{ \log n}{\log (d(G)/k)-c_2}\rceil$. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Tur\'an number of $C^r_{2k}$ with improved coefficients. Furthermore, we show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_3=c_3(r)$ and $c_4=c_4(r)$ such that every $n$-vertex linear $r$-uniform graph with average degree $d(G)\geq c_3k$, contains linear cycles of $k$ consecutive lengths, the shortest of which has length at most $6\lceil \frac{\log n}{\log (d(G)/k)-c_4} \rceil +6$. Both the degree condition and the shortest length among the cycles guaranteed are best possible up to a constant factor.