Lower bounds for the number of small convex k-holes ?
Let S be a set of n points in the plane in general position, that is, no three points of S  are on a line. We consider an Erd?s-type question on the least number hk(n) h k ( n ) of convex k-holes in S , and give improved lower bounds on hk(n) h k ( n ) , for 3?k?5 3 ? k ? 5 . Specifically, we show that h 3 ( n ) ? n 2 ? 32 n 7 + 22 7 , h 4 ( n ) ? n 2 2 ? 9 n 4 ? o ( n ) , and h 5 ( n ) ? 3 n 4 ? o ( n ) . We further settle several questions on sets of 12 points posed by Dehnhardt in 1987.
empty convex polygon erd?s-type problem counting