E. Nevo ; T. K. Petersen

On $\gamma$vectors satisfying the KruskalKatona inequalities
dmtcs:2842 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

https://doi.org/10.46298/dmtcs.2842
On $\gamma$vectors satisfying the KruskalKatona inequalities
Authors: E. Nevo ^{1}; T. K. Petersen ^{2}
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E. Nevo;T. K. Petersen
1 Department of Mathematics [Cornell]
2 DePaul University [Chicago]
We present examples of flag homology spheres whose $\gamma$vectors satisfy the KruskalKatona inequalities. This includes several families of wellstudied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose $f$vectors are the $\gamma$vectors in question, and so a result of Frohmader shows that the $\gamma$vectors satisfy not only the KruskalKatona inequalities but also the stronger FranklFürediKalai inequalities. In another direction, we show that if a flag $(d1)$sphere has at most $2d+3$ vertices its $\gamma$vector satisfies the FranklFürediKalai inequalities. We conjecture that if $\Delta$ is a flag homology sphere then $\gamma (\Delta)$ satisfies the KruskalKatona, and further, the FranklFürediKalai inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $\gamma$vectors are nonnegative.