On Erdős–Moser Problems

Abstract

In this thesis, we introduce a problem in number theory posed by Erdős and Moser in 1965, and a solution provided by Stanley who used a technique in algebraic geometry.

Erdős–Moser Problem is about the sum of subsets of finite set. More precisely, let S⊂R be a finite set. For any α∈R, define f(S,α)=|{T⊂S : Σ_{t∈T}{t}=α}|. Erdős–Moser Problem asks how large f(S,α) can be if we require |S|=n. Erdős–Moser Conjecture that was introduced in 1965, asserts that if |S|=2m+1, then f(S,α)≤ f({-m,-m+1,...,m},0). Stanley proved the conjecture in1980: Let S be a set of 2l+1 distinct real numbers and let T_{1},...,T_{k} be subsets of S whose element sums are equal. Then k does not exceed the middle coefficient of the polynomial 2(1+q)²(1+q²)²…(1+q^{l})², and this bound is best possible. We study Stanley's results including variations of Erdős–Moser Problems. Then we consider another variation of Erdős–Moser Problem by restricting the number of elements and give a solution to the problem by applying Stanley's result.