Extremal properties of regular and affine generalized m-gons as tactical configurations

The purpose of this paper is to derive bounds on the sizes of tactical configurations of large girth which provide analogues to the well-known bounds on the sizes of graphs of large girth. Let exα(v,g) denote the greatest number of edges in a tactical configuration of order v, bidegree a, aα and girth at least g. We establish the upper bound exα(v,g)=O(v1+1τ), where τ=14(α+1)g−1 for g≡0(mod4) and τ=14(α+1)g+12(α−3) for g≡2(mod4). We further demonstrate this bound to be sharp for the regular and affine generalized m-gons but not for the nonregular generalized m-gons. Finally, we derive lower bounds on exα(v,g) via explicit group theoretic constructions.