Eisenbud–Green–Harris conjecture for homogeneous ideals containing regular sequences

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, and let $I\subseteq S$ be a homogeneous ideal containing a regular sequence $f_1,\dots,f_n$ with degrees

$$2\leq a_1=\deg(f_1)\leq\cdots\leq a_n=\deg(f_n).$$

Eisenbud–Green–Harris conjecture. The ideal $I$ has the same Hilbert function as an ideal containing $x_1^{a_1},\dots,x_n^{a_n}$. This conjecture predicts that the Hilbert function of an ideal containing a regular sequence is attained by an ideal containing the corresponding pure powers; the paper studies this conjecture for ideals generated by quadrics and proves it for almost complete intersections of quadrics in six variables.