In mathematics, the Erdős–Szekeres theorem is a finitary result that makes precise one of the corollaries of Ramsey's theorem. While Ramsey's theorem makes it easy to prove that every sequence of distinct real numbers contains a monotonically increasing infinite sub sequence or a monotonically decreasing infinite sub sequence, the result proved by Paul Erdős and George Szekeres goes further. For given r, s they showed that any sequence of length at least (r − 1)(s − 1) + 1 contains a monotonically increasing sub sequence of length r or a monotonically decreasing sub sequence of length s. The proof appeared in the same 1935 paper that mentions the Happy Ending problem.So I'm writting a paper on Paul Erdos, but im still early on in my math undergrad. Im trying to find a simple answer as to what Erdos did with Ramsey theory and extremal combinatorics, but i keep coming across tough to understand scientific papers.