The initial ideal of generic sequences and Fr\"{o}berg's Conjecture

Let $K$ be an infinite field and let $I=(f_1,\cdots ,f_r)$ be an ideal in the polynomial ring $R=K[x_1,\cdots ,x_n]$ generated by generic forms of degrees $d_1,\cdots ,d_r$. A longstanding conjecture by Fr"{o}berg predicts the shape of the Hilbert function of $R\mathrm{/}I\mathrm{.}$ In 2010 Pardue stated a conjecture on the initial ideal of $n$ generic forms with respect to the deg-revlex order and he proved that it is equivalent to Fr"{o}berg’s Conjecture. We study Pardue’s Conjecture and we prove it under suitable conditions on the degrees of the forms. This yields a partial solution to Fr"{o}berg’s Conjecture in the case $r\le n+2$ over an infinite field of any characteristic.