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...
