Manickam-Mikl\'os-Singhi Conjectures on Partial Geometries

TL;DR

This paper proves the Manickam-Miklós-Singhi conjecture for certain partial geometries by establishing conditions under which it holds and identifies specific counterexamples.

Contribution

It provides a condition on partial geometries that guarantees the MMS conjecture and describes particular geometries that serve as counterexamples.

Findings

MMS conjecture holds under specific conditions on partial geometries

Identification of particular partial geometries that counter the conjecture

Extension of the conjecture's applicability to new geometric structures

Abstract

In this paper we give a proof of the Manickam-Mikl\'os-Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counter-examples to the conjecture are described.