Update (Sept. 28): Yuval Wigderson has made a further improvement on the multicolor Ramsey number bound for more than three colors.
The Ramsey number r(t; ℓ) is the smallest natural number n such that every ℓ-coloring of the edges of the complete graph contains a monochromatic . (r(t;2) is often denoted by R(t,t) and r(t;3) by R(t,t,t) etc.) Famously, R(3,3)=6; R(4,4)=18; and R(3,3,3)=17. Understanding R(t,t) is among the most famous problems in combinatorics. (Understanding if r(3; ℓ) is exponential or superexponential in ℓ is also a very famous problem.)
It is known since the 1940s that .
Lower bounds for R(t,t) where used by Lefmann in 1987 to give lower bounds on r(t; ℓ) for ℓ>2. Asaf Ferber and David Conlon gave now exponential improvement. This is truly remarkable and the paper is just 4-page long! congratulations Asaf and David!
Expect more cheering news of discrete geometry nature from Oberwolfach. (I take part remotely in the traditional meeting on Discrete and computational geometry, see pictures below).
Update (Sept 24.): An excellent blog post on Anurag math blog. Anurag describes in details the construction, describes the connections with finite geometries, and improves the construction to get a better result. (See also there many cool posts, e.g., an earlier post with some connections of finite geometries and different Ramsey problems Heisenberg groups, irreducible cubics and minimal Ramsey.)