Ramsey property for spaces with bilinear forms

TL;DR

The paper investigates the Ramsey property for vector spaces over finite fields endowed with bilinear forms, proving that symplectic spaces over a finite field (notably the two-element field) do not possess the Ramsey property. It also analyzes spaces with skew-symmetric forms and radicals of finite codimension, showing the Ramsey property can fail in these cases as well. Methodologically, the authors employ Fraïssé theory, colorings of subspace copies, Witt's theorem for isometries, and projections to hyperbolic parts to derive negative results, while demonstrating that introducing a fixed finite distinguished subspace can restore Ramsey properties for families of subspaces containing it. The results connect to generalized affine spaces and raise questions about finite Ramsey degrees in this setting, outlining both obstructions and potential structure that yields positive Ramsey outcomes under additional constraints.

Abstract

We study the Ramsey property for vector spaces over finite fields with bilinear forms. We prove that symplectic spaces over finite fields do not have the Ramsey property. We also describe vector spaces with skew symmetric bilinear forms and radicals of finite codimension, where the Ramsey property does not hold. Some direct connections with generalized affine spaces are given.

Theorems & Definitions (12)

• Remark 1
• Remark 1.1
• Lemma 1.2
• Theorem 1.3
• Remark 2.1
• Lemma 2.2
• Theorem 2.3
• Proposition 2.4
• Proposition 2.5
• Corollary 2.6