Linear independence over naturally-ordered semirings with applications to dimension arguments in extremal combinatorics

TL;DR

This paper introduces a new algebraic framework using difference ordered semirings to analyze linear independence, enabling dimension-based combinatorial arguments and solutions to problems involving the disparate union property.

Contribution

It develops a novel notion of linear independence over a class of semirings and applies it to extremal combinatorics, extending dimension arguments beyond traditional vector spaces.

Findings

Maximal size of families with the disparate union property characterized

Linear independence over certain semirings can be efficiently detected via bideterminant

Dimension arguments extended to new algebraic structures in combinatorics

Abstract

A family of subsets $\mathcal{F} \subseteq \mathcal{P}(\{1, 2, \ldots, n\})$ has the disparate union property if any two disjoint subfamilies $\mathcal{F}_1, \mathcal{F}_2 \subseteq \mathcal{F}$ have distinct unions $\bigcup \mathcal{F}_1 \neq \bigcup \mathcal{F}_2$; what is the maximal size of a family with the disparate union property? Is there a simple and efficiently computable characterization of size-maximal families? This paper highlights a class of partially-ordered semirings -- difference ordered semirings with a multiplicatively absorbing element -- and shows it is common and easily constructed. We prove that a suitably modified definition of linear independence for semimodules over such semirings enjoys the same maximality property as for vector spaces, and can furthermore be efficiently detected by the bideterminant. These properties allow us to extend dimension argument in extremal combinatorics and provide simple and direct solutions to the puzzles above.