consider the set of graphs with $n$ vertices and exactly half of all $\binom n 2$ possible edges.
looking for a formula that counts the number of these graphs that have a $k$-clique but not a $(k+1)$-clique.
looked at some of the Erdos-Renyi random graph theory and related formulas but did not see this case covered so far. an estimate may be ok. also if this is used in a paper somewhere, that would be useful to know.
edit as Erdos-Renyi theory & a comment points out the critical point for detection of a $k$-clique is at $k=\log(n)$ where the probability goes from $P<0.5$ to $P>0.5$. it would be very interesting if there was a formula that could be derived independent of these regions (once called "subcritical, critical, supercritical"), but am seeking the answer for $k \approx \log(n)$ in particular.
background/motivation: question inspired by similar constructions in theoretical computer science circuit theory proofs/theorems.