Ramsey numbers of semi-algebraic and semi-linear hypergraphs

An r-uniform hypergraph H is semi-algebraic of complexity t=(d,D,m) if the vertices correspond to points in Rd and edges are determined by sign-pattern m degree-D polynomials. Semi-algebraic hypergraphs bounded provide a general framework for studying geometrically defined hypergraphs. The much-studied Ramsey number Rrt(s,n) denotes smallest N such that every t on contains either clique size s or an independent set n. Conlon, Fox, Pach, Sudakov Suk proved Rrt(n,n)n(log⁡n)1/3−o(1) some t. In addition, motivated results Bukh Matoušek Basit, Chernikov, Starchenko, Tao Tran, study problem when defining polynomials linear, is, D=1. particular, prove Rrd,1,m(n,n)≤2O(n4r2m2), while from below, establish Rr1,1,1(n,n)≥2Ω(n⌊r/2⌋−1).