3 problems

Homological connectivity threshold conjecture.

Let $d \geq 1$ and $n \geq 4$ be integers. Suppose $U_1,\ldots,U_n$ are independent random $d$-vectors uniformly distributed over the cube $C_n:= [0,n^{1/d})^d$, and let $V$ be the…

Let $p(n,\alpha)$ be the probability that $n$ spherical caps of angular radius $\alpha$ with independently and uniformly chosen centers do not cover the unit sphere $S^2$. Let…