Relationships between lazy burning numbers of design hypergraphs and design dimension

Determine whether there exist relationships between the lazy burning number b_L(H) of hypergraphs arising from combinatorial designs other than Steiner Triple Systems and an appropriate notion of dimension for those designs, specifically investigating hypergraphs constructed from balanced incomplete block designs (BIBDs).

Background

The paper studies lazy burning in hypergraphs and provides several characterizations and complexity results. It also notes a known relationship for Steiner Triple Systems: the lazy burning number of the associated hypergraph equals the design’s dimension.

Motivated by this, the authors ask whether analogous relationships hold for other combinatorial designs, with balanced incomplete block designs (BIBDs) highlighted as a primary example. Establishing such connections would generalize the Steiner Triple System result and could link burning dynamics to structural design parameters.