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

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.

💡 Research Summary

The paper extends the classical Erdős‑Ko‑Rado (EKR) theorem to the setting of projective Hjelmslev geometries built over finite chain rings whose nilpotency index is two. After recalling the algebraic background on finite chain rings, their modules, and the construction of the projective Hjelmslev geometry PHG(n‑1,R), the authors define τ‑intersecting families of subspaces: a family F of subspaces of a fixed shape κ is τ‑intersecting if any two members intersect in a subspace of shape τ. Here κ and τ are expressed as multiples of the chain‑ring length m, i.e., κ=m·k and τ=m·t with 1≤t<k≤n/2.

A key technical tool is the family of residue maps η_i : R → R/N^i, which project subspaces of PHG(n‑1,R) onto subspaces of the classical projective space PG(n‑1,q). Lemma 3.1 shows that the image η_i(F) of a τ‑intersecting family is a τ′‑intersecting family in PG(n‑1,q) with appropriately reduced parameters. This reduction allows the authors to apply known q‑analogues of the EKR theorem (including Tanaka’s result) to the projected families.

The main results are Theorem 3.2 and Theorem 3.3. Theorem 3.2 deals with τ=m·t, κ=m·k families that avoid a certain “neighbour class”