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 $\Delta$ is $2(i+j)$-Cohen--Macaulay for all $0\leq j\leq \dim\Delta+1-i$, and (2) for every $i$-banner simplicial complex $\Delta$ there exists a balanced complex $\Gamma$ with the same number of vertices as $\Delta$ whose face numbers of dimension $i-1$ and higher coincide with those of $\Delta$. 
Steven Klee and Isabella Novik
math.CO