graph theory - How to estimate the growth of the probability that $G(n, M)$ contains a $k$-clique

You might take a look at Chapter VII of Bollobas. In particular,
Theorem VII.1.7 -- which is simple enough that he doesn't bother providing a proof -- states that the expected number of $k$-cliques in $G(n,M)$ is, setting $N={n choose 2}$ and $K={k choose 2}$,

$${n choose k} {n-K choose M- K} {N choose M}^{-1}.$$
Also, Theorem VII.3.7 states that if $M=o(n^{-2/(k-1)})$ then with probability tending to one, $G_{n,M}$ contains no $k$-clique, whereas if $M/n^{-2/(k-1)} to infty$ then with probability tending to one $G_{n,M}$ does contain a $k$-clique. I know this doesn't fully answer your question but it may help.

Incidentally, (you probably already realize that) it is a priori possible (though I don't think it is the case) that, for example, $t_k(M+1)-t_k(M) geq frac{1}{mathrm{poly}(n)}$ for all ${k choose 2}leq M leq lceil frac{(k-1)N}{k}rceil$, since all we really know by Turán is that

$$sum_{M=K}^{lceil(k-1)N/krceil} (t_k(M+1)-t_k(M)) = 1.$$

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