Given a positive integer $a$, the *Ramsey number* $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color.

For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know each other, or 3 that don't know one another; but there is a party of 5 people without this property. This is probably known to everybody.

Slightly less known is the fact that any such coloring of $K_6$ in fact contains **2** monochromatic triangles.

The *Ramsey multiplicity* $m(a)$ (there does not seem to be a standard notation) is the largest number $m$ of monochromatic copies of $K_a$ that we can guarantee in any 2-coloring of $K_{R(a)}$. For example, $m(3)=2$, and Piwakowski and Radziszowski showed around 1999 that $m(4)=9$.

I have a couple of questions (please forgive me if they are trivial, I'm just beginning to form my intuitions in this field) :

Is it known that $m(n)$ is monotonically increasing?

Do we know anything about the rate of growth of the function $m(n)$?

I suspect that the answer to both questions is yes and that reasonable bounds for $m(n)$ are known, but haven't been able to locate any references. The best I know is that $$ m(n)\le \frac{\binom{r(n)}n}{2^{\binom n2-1}}, $$
(proved by Burr and Rosta in 1980), which is probably too high, and a recent result of Conlon suggests that
$$ m(n)\ge C\frac{\binom{r(n)}n}{2^{n(3n-1)/2}} $$
for some appropriate $C$. I say "suggests" because Conlon's results carry some additional implicit constants that I haven't checked can be absorbed this way. (Please let me know if I am completely off the mark here.) [**Edit:** Unfortunately, Conlon's bounds (in his paper "On the Ramsey Multiplicity of Complete Graphs") do not apply here. No lower bound beyond $m(n)\ge1$ seems known.]

"Update": William Gasarch's Open Problems column in the June 2020 issue of the *ACM SIGACT News* is devoted to Ramsey multiplicity.

bianglewhen a red edge and a blue edge meet at a vertex. There are at most six biangles at each vertex of $K_6$, so at most $36$ biangles total in a coloring of $K_6$. Each non-monochrome triangle has exactly two biangles, so there are at most $36/2=18$ non-monochrome triangles. But there are ${6\choose3}=20$ triangles in $K_6$, so at least two of them must be monochrome. $\endgroup$5more comments