Connections in Extremal Combinatorics

Extremal combinatorics is that part of discrete mathematics that studies how large or small a collection of finite objects can be under given restrictions and has broad connections with number theory, discrete geometry, probability, theoretical computer science and beyond. Recent advances in this area have brought to the fore several unexpected connections between seemingly disparate problems. In this project, the PI will explore some of these connections further to resolve several old problems in the area and forge further connections with other areas. The research will involve graduate students and postdocs. Of particular interest are the recent breakthroughs in graph Ramsey theory and the advance of Mattheus and Verstraëte specifically, which has revealed intriguing connections between the study of off-diagonal Ramsey numbers and problems in extremal graph theory and finite geometry. The PI also intend to build on previous progress by the PI and his collaborators to make further progress in graph Ramsey theory, extremal graph theory and the fundamental areas of additive combinatorics and discrete geometry. In doing so, the PI expects to open further connections and increase interactions between extremal combinatorics and important recent trends in the study of high-dimensional expansion, convex geometry and algebraic combinatorics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.