CIIM 2020 Problem 3

Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$ . Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$ , there exist at least $t$ elements of $A_i$ , such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$ . Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$ .