From Flag Complexes to Banner Complexes

A notion of an i-banner simplicial complex is introduced. For various values of i, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary dimension that are (i+1)-banner but not i-banner are constructed. It is shown that several theorems for flag complexes have appropriate i-banner analogues. Among them are (1) the codimension-(i + j − 1) skeleton of an i-banner homology sphere ∆ is 2(i + j)-Cohen–Macaulay for all 0 ≤ j ≤ dim∆ + 1− i, and (2) for every i-banner simplicial complex ∆ there exists a balanced complex Γ with the same number of vertices as ∆ whose face numbers of dimension i− 1 and higher coincide with those of ∆.