Bounds for cops and robber pursuit

We prove that the robber can evade (that is, stay at least unit distance from) at least ⌊n/5.889⌋ cops patroling an n×n continuous square region, that a robber can always evade a single cop patroling a square with side length 4 or larger, and that a single cop on patrol can always capture the robber in a square with side length smaller than 2.189… .