A conjecture of recski in combinatorial set theory

A p-cover of $\text{n = {1, 2,…,}\text{n}\text{}}$ is a family of subsets $\text{S}{}_{\mathrm{i}}\text{≠ ?}$ such that ∪ S_i = n and |S_i ∩ S_i| ? p for i ≠ j. We prove that for fixed p, the number of p-cover of n is O(n^p+1logn).