On the degree two entry of a Gorenstein h-vector and a conjecture of Stanley

In this short paper we establish a (non-trivial) lower bound on the degree two entry h₂ of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1,r,h₂,r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h₂ may assume. In fact, we show that lim _r→∞f(r)/r^2/3=6^2/3. In general, we wonder whether our lower bound is sharp for all integers e ≥ 4 and r ≥ 2.