TITLE: What is the chromatic number of the "conic hypergraph" on a non-singular plane cubic?
QUESTION [11 upvotes]: Can we color the points of a complex non-singular plane cubic curve with finitely many colors so that no conic intersects the curve at 6 distinct points of the same color?
If so, what is the smallest number of such colors?

REPLY [9 votes]: You mean the six points to be distinct, of course (or not all six points to be the same point).
Fixing the analytic identification $(\wp(z),\wp'(z))$ with $T = \mathbb{C}/\mathbb{\Lambda}$, the Abel-Jacobi theorem implies that the six points of intersection of $E \subset \mathbb{CP}^2$ with any conic sum to zero on the torus. Now take the fundamental parallelogram $P$ of $\Lambda$, subdivide it into $36$ equal subparallelograms, and use $72$ colors so that monochromatic subsets of $P$ are either interior points of one of the subparallelograms or of its left or down boundary edges, or else a single lattice point from the subdivision of $P$ (a $6$-division point).