Jun 5, 2016 | Atlanta, GA
Through a series of research papers posted online in recent weeks, mathematicians have solved a problem about the pattern-matching card game, Set, that predates the game itself. The proof, whose simplicity has stunned mathematicians, is leading to advances in other combinatorics problems.
Three mathematicians in particular, Ernie Croot of the Georgia Institute of Technology (pictured to the right), Vsevolod Lev of the University of Haifa, Oranim, in Israel, and Péter Pál Pach of the Budapest University of Technology and Economics in Hungary, posted a paper online, on May 5, showing how to use a polynomial method to solve a closely related problem. In their work, the three researchers used Set attributes with four different options instead of three. For technical reasons, this problem is more tractable than the original Set problem. Not long after this, two mathematicians, Jordan Ellenberg, and Dion Gijswijt, each independently posted papers showing how to modify the argument to polish off the original cap set problem, and a joint paper combining their results.
The work of Ernie Croot and his collaborators is continuing to make huge waves, with many interesting consequences now unfolding. Their work has already been applied to matrix multiplication, tri-colored sum-free sets, and the Erdös-Szemerédi sunflower conjecture, which concerns sets that overlap in a sunflower pattern. Their work was also featured in an article in Quanta Magazine, which gives a more detailed history of this recent breakthrough.