Abstract: We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $\Delta$, we show that there exists a balanced complex with the same $f$-vector as $\Delta$, and whose top-dimensional Betti number is at least that of $\Delta$, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of $\Delta$ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bo...
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Topics: 
Combinatorics
Discrete mathematics
Pure mathematics