A high girth graph construction and a lower bound for hitting set size for combinatorial rectangles

We give the following two results. First, we give a deterministic algorithm which constructs a graph of girth $log_k(n)+O(1)$ and minimum degree k-1, taking number of nodes n and the number of edges e=[nk/2] as input. The graphs constructed by our algorithm are expanders of sub-linear sized subsets, that is subsets of size at most $n^\delta$, where $\delta$