Let $k \geq 3$ be fixed.  Ramsey's theorem says that if $n$ is sufficiently large and we color the edges of the complete graph $K_n$ red or blue, there must be at least one monochromatic $K_k$.  As it turns out, it's not just "at least one" but many: An averaging argument shows that as $n \rightarrow \infty$ a positive fraction of all the $\binom{n}{k}$ copies of $K_k$ in our coloring must be monochromatic.  

This leads to a natural follow-up question: How few copies can we get?  If we consider all $2$-colorings of $K_n$, which one (asymptotically) minimizes the number of monochromatic copies of $K_k$?  

This was first studied for the case $k=3$ (monochromatic triangles) by  [Goodman, who in 1959][1] gave an explicit answer asymptotic to $\frac{1}{4} \binom{n}{3}$.  The fraction $\frac{1}{4}$ has a natural interpretation here -- if we color *randomly*, this is the expected fraction of monochromatic triangles.  Three years later, [Erdős observed][2] that the random coloring gives an upper bound of $2^{-\binom{k}{2}+1}$ on the minimum fraction of monochromatic $K_k$, and said it "seems likely" this was asymptotically optimal.

By 1980 [Burr and Rosta conjectured][3] that something even stronger was true: For *any* fixed graph $H$ the asymptotic way to minimize monochromatic copies of $H$ was just to color randomly.  It wasn't until 1989 that [Sidorenko gave a counterexample to the Burr-Rosta conjecture][4] (a triangle with a pendant edge) and [Thomason disproved Erdős's original conjecture][5] by giving a coloring with significantly fewer monochromatic $K_4$ then random.  

It is still an open question to determine the optimal coloring to minimize monochromatic $K_4$, and also still an open question to determine for which graphs the Burr-Rosta conjecture is true (such graphs are termed "common" in the literature).   


  [1]: https://www.jstor.org/stable/2310464
  [2]: https://users.renyi.hu/~p_erdos/1962-14.pdf "On the number of complete subgraphs contained in certain graphs"
  [3]: https://onlinelibrary.wiley.com/doi/pdf/10.1002/jgt.3190040403
  [4]: https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=207598&sort=Newest&vfpref=html&r=28&mx-pid=1033422
  [5]: https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=172220&sort=Newest&vfpref=html&r=57&mx-pid=991659