Kühn­el–Kalai conjecture on vertex bounds for even-dimensional manifolds

Let $M$ be a combinatorial $2k$-manifold with $n$ vertices, Euler characteristic $c9(M)$, and let a triangulation be $(k+1)$-neighborly when every set of at most $k+1$ vertices spans a simplex.

Kühn­el–Kalai conjecture.

$$\binom{n-k-2}{k+1}\geq (-1)^k\binom{2k+1}{k+1}\bigl(\chi(M)-2\bigr),$$

with equality if and only if the triangulation is $(k+1)$-neighborly.

This conjecture generalizes the Heawood bound and the four-dimensional analogue proved by Kühn­el. Its general validity and the equality characterization remain open.