The classical Ramsey numbers r(s,t) denote the minimum n such that every red-blue coloring of the edges of the complete graph K_n contains either a red clique of order s or a blue clique of order t. These quantities are the centerpiece of graph Ramsey Theory, and have been studied for almost a century. The Erdős-Szekeres Theorem (1935) shows that for each s ≥ 2, r(s,t) = O(t^s - 1) as t →∞. We introduce a new approach using pseudorandom graphs which shows r(4,t) = Ω(t^3/(log t)^4) as t →∞, answering an old conjecture of Erdős, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems.
A graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n\}$ is an intersection graph of segments if there are segments $s_1,\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\{v_i,v_j\}$ is an edge of~$G$. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments. It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by $\exists\mathbb{R}$. This class consists of problems that can be reduced, in polynomial time, to solvability of a system of polynomial inequalities in several variables over the reals. We discuss some subtleties in the definition of $\exists\mathbb{R}$, and we provide a complete and streamlined account of a proof of the $\exists\mathbb{R}$-completeness of the recognition problem for segment intersection graphs. Along the way, we establish $\exists\mathbb{R}$-completeness of several other problems. We also present a decision algorithm, due to Muchnik, for the first-order theory of the reals.
