The relevance of Freiman's theorem for combinatorial commutative algebra

TL;DR

This paper explores the application of Freiman's theorem to monomial ideals and fiber cones, providing algebraic characterizations and classifying related graph structures.

Contribution

It introduces the concept of Freiman ideals in monomial ideals and characterizes them algebraically, linking combinatorial bounds to algebraic structures.

Findings

Characterization of Freiman ideals in monomial ideals

Classification of graphs with Freiman edge ideals

Connection between Freiman's theorem and algebraic properties

Abstract

Freiman's theorem gives a lower bound for the cardinality of the doubling of a finite set in $\RR^n$. In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman's lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.