2 problems

Let $\Delta$ be a flag $4$-sphere with $n$ vertices. Its number of edges is $f_1(\Delta)$. Adam–Hladký's conjecture. Every flag $4$-sphere with $n$ vertices satisfies … This would…

Let $d$ and $k$ be integers with $d\neq 2k+1$, and let ${\cal K}_k(d)$ denote the generalized Walkup class of $d$-dimensional simplicial complexes. A simplicial complex is $(k+1)$-…