Counting monochromatic copies of K_4: a new lower bound for the Ramsey   multiplicity problem
  Denote by k_4(n) the minimal number of monochromatic copies of a K_4 in a 2-colouring of the edges of K_n and let c_4 := lim k_4(n)/\binom{n}{4}. The best known bounds so far were given by Thomason, who proved that c_4 < 1/33 \approx 0.0303, and Giraud, who showed that c_4 > 1/46 \approx 0.0217. In this paper we prove the new lower bound c_4 > 204603019 / 7112448000 > 0.0287. 
Susanne Nie{\ss}
math.CO