3 problems

Wedge-of-spheres conjecture. For all $k\geq 2$, the complex $\mathsf{Cliq}_{k}(G)$ is homotopy equivalent to a wedge sum of $(2k-3)$-dimensional spheres. Moreover, the number of sp…

Let $X(n,p)$ be the clique complex of the Erdős–Rényi random graph $G(n,p)$, and fix an integer $k\geq 0$. A random property holds almost always if its probability tends to $1$ as…

Let $G$ be a graph and let $k$ be a nonnegative integer. Consider the serial algorithm and parallel algorithm for constructing the $k$-skeleta of the clique complex of $G$ describe…