The Eisenbud-Green-Harris conjecture for fast-growing degree sequences

Let $S$ be a standard graded polynomial ring over field, and $I$ homogeneous ideal that contains regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when forms satisfy $d_i \geqslant \sum_{j=1}^{i-1}(d_j-1)$, improving result obtained in 2008 by first author Maclagan. Except for sporadic case five quadrics, recently proved Gunturkun Hochster, results this article recover all known cases where only are fixed, include several additional ones.