Extremal Betti Numbers and Persistence in Flag Complexes

We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on n vertices, we show that βk(X(G)) is maximal when G = Tn,k+1, the Turán graph on k + 1 partition classes, where X(G) denotes the flag complex of G. Building on this, we construct an edgewise (one edge at a time) filtration G = G1 ⊆ · · · ⊆ Tn,k+1 for which βk(X(Gi)) is maximal for all graphs on n vertices and i edges. Moreover, the persistence barcode Bk(X(G)) achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with|E(Tn,k+1)| edges. For k = 1, we consider edgewise filtrations of the complete graph Kn. We show that the maximal number of intervals in the persistence barcode is obtained precisely when G⌈n/2⌉·⌊n/2⌋ = Tn,2. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize β1(X(Gi)) for all i, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of Kn.