A Chain Ring Analogue of the Erdos-Ko-Rado Theorem

TL;DR

The paper extends the Erdős–Ko–Rado paradigm to intersecting families of subspaces in projective Hjelmslev geometries over finite chain rings with nilpotency index $m=2$, proving a tight EKR-type bound for tau-intersecting families of fixed shape and describing when equality holds. Central to the approach are the preliminaries on finite chain rings, their module shapes, and the nested PHG structure, together with an induction on the length of the chain and projections $\eta_i$ to factor geometries. A key contribution is a Tanaka-style bound for shapes $\kappa=m^k$ and intersections of at least $t$-dimensional subspaces, along with a classification of the extremal configurations. The paper also presents a non-canonical, maximal EKR-family of non-free subspaces, demonstrating that maximal intersecting families need not be canonically intersecting even in the Hjelmslev setting, and provides an explicit construction showing the richness of these geometries for combinatorial applications.

Abstract

In this paper, we prove an analogue of the Erdős-Ko-Rado theorem intersecting families of subspaces in projective Hjelmslev geometries over finite chain rings of nilpotency index 2. We give an example of maximal families that are not canonically intersectng.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Lemma 3.1