3 problems

Let $r,d$ be positive integers. Given $d+1$ sets $X_1, \ldots, X_{d+1}$, each consisting of $r$ points of $\mathds{R}^d$, consider partitions of their union into $r$ sets…

Let $r,d$ be fixed positive integers, let $N$ be a positive integer, and let $p_r$ be the probability that a random permutation of $r$ numbers has at least one fixed point. For $N$…

Let $r$ be a positive integer. Given $r$ blue, $r$ red, and $r$ green lines in the plane in general position, where a family is in general position when the normal vectors of any t…