We study how Betti numbers of ideals in a local ring change under small perturbations. Given p ? N and given an ideal I Noetherian ( R , m ) our main result states that there exists > 0 such if J is with ? mod the same Hilbert function as then ? i / coincide for ? . Moreover, we present several cases which forced to have therefore numbers.

This paper describes an application of research that sits at the intersection of commutative algebra and combinatorics: Betti numbers of graphs. In particular, we describe a correspondence between simple undirected graphs and a class of ideals in a polynomial ring. We then briefly introduce some of the algebraic invariants that can be associated to the ideal and the relation of these invariants...