Flag complexes and homology

Kai Fong Ernest Chong, Eran Nevo

Research output: Contribution to journalArticlepeer-review


We prove several relations on the f-vectors and Betti numbers of flag complexes. For every flag complex Δ, we show that there exists a balanced complex with the same f-vector as Δ, and whose top-dimensional Betti number is at least that of Δ, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of Δ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the f-vector of Δ, in terms of the top-dimensional Betti number of Δ. This result has a continuous analog: If Δ is a (d−1)-dimensional flag complex whose (d−1)-th reduced homology group has dimension a≥0 (over some field), then the f-polynomial of Δ satisfies the coefficient-wise inequality fΔ(x)≥(1+(ad+1)x)d.

Original languageEnglish
Article number105466
JournalJournal of Combinatorial Theory. Series A
StatePublished - 1 Aug 2021
Externally publishedYes


  • Betti number
  • Clique complex
  • Flag complex
  • Homology
  • Turán graph
  • f-vector

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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