Infinitely many polygons, no four have a common point
Solution 1:
Suppose there are $n$ $2015$-gons. Divide each $2015$-gon into $2013$ triangles. Then since any three $2015$-gons share a common point, then there exist three triangles which share a common point, yielding at least $\binom{n}{3}$ triples of triangles sharing a common point.
Now, by Fractional Helly's Theorem, we are done since we can choose $n$ to be large enough such that $4$ of these triangles must share a common point.