Eigenvalues and cycles of consecutive lengths

As the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollob\'as in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum $C$ such that for all positive $\varepsilon \sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for each integer $\ell\in[3,(C-\varepsilon)n]$. We prove that $C\geq\frac{1}{4}$ by a novel method, improving the existing bounds. Besides several novel ideas, our proof technique is partly inspirited by the recent research on Ramsey numbers of star versus large even cycles due to Allen, {\L}uczak, Polcyn and Zhang, and with aid of a powerful spectral inequality. We also derive an Erd\H{o}s-Gallai-type edge number condition for even cycles, which may be of independent interest.