A spectral condition for cycles with consecutive lengths: a Ramsey Theory approach

Nikiforov posed the following open problem: What is the maximum $C$ such that for all positive $\epsilon<C$ and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius $\rho(G)>\sqrt{\lfloor n^2/4 \rfloor}$ contains a cycle of length $l$ for each integer $l\in [3,(C-\epsilon)n]$. This can be seen as a spectral version of a classical theorem in extremal graph theory, which says that any graph $G$ contains all cycles $C_l$ for each $l\in [3,\lfloor (n+3)/2 \rfloor]$ if $e(G)>n^2/4$. We prove that $C\geq 1/4$ by a novel method, which improves the existing bounds.