Miklós–Manickam–Singhi conjectures on partial geometries

Abstract

In this paper we give a proof of the Manickam-Miklós-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 counterexamples to the conjecture are described.

A new partial geometry pg(5,5,2)

“…A stronger theorem along the same lines was proved in [8,Corollary 52]. Furthermore, in [5] the geometry of van Lint and Schrijver was listed as a counterexample to the Manickam-Miklós-Singhi conjecture for partial geometries.…”

“…A partial geometry pg(5, 5, 2) has v = 81 points and as many lines. We shall denote the geometry of van Lint and Schrijver by G = (P, L), where the set of lines L consists of 6-element subsets of the set of points P. Two constructions of G are given in [11], the first using cyclotomy in the finite field F 81 , and the second using the dual code of the repetition code in F 5 3 . The first construction does not essentially use multiplication in F 81 and we describe it here in purely linear algebraic terms.…”

“…Negative lines were used in [5] to show that G does not have the strict MMS star property. There are 27 cliques of size 6 in the line graph Γ 2 (G ′ ) that are not stars.…”

“…Van Lint and Schrijver [11] constructed a partial geometry with parameters pg (5,5,2). Another construction of the same partial geometry was given in [1].…”