Combinatorics Seminar

A system of linear equations L is common over F_p if any 2-coloring of F_p^n gives at least as many monochromatic solutions to it as a random 2-coloring, asymptotically as n->infty. When L is a single equation, Fox, Pham and Zhao gave a complete characterization of common linear equations. When L consists of two equations, Kamčev, Liebenau and Morrison showed that all irredundant 2*4 linear systems are uncommon. In joint work with Anqi Li and Yufei Zhao, we: (1) determine commonness of all 2*5 linear systems up to a small number of cases; (2) show that all 2*k linear systems with k even and girth (length of the shortest equation) k-1 are uncommon, answering a question of Kamčev, Liebenau and Morrison.