Istvan Reiman's theorem gives an upper bound of $\left \lfloor \dfrac {n} {4} (1+\sqrt{4n-3})\right\rfloor$ for $n=8$, amounting to $12$ edges, but it cannot be realized; a (symmetric) model with $11$ edges is easy to find: vertices $\{a,b,c, p,q,r, x,y\}$ and edges $\{ab,bc,ca, pq,qr,rp, ap, xb,xq,yc,yr\}$.

What is Istvan Reiman's theorem?

It is what I said it was; it gives that expression as an upper bound for the number of edges of a simple graph on $n$ vertices containing no cycle of length $4$.