Joints Problem

Given a set of lines in R^3, a point is said to be a joint if there are three non-coplanar lines passing through the point. The joints problem asks what the maximum number of joints is with N lines.

Lower Bound
Consider M planes in R^3 in generic positions. Our lines would be intersections of two planes, so in total we have M(M-1)/2 lines. If we look at the three lines that come from three of our planes (the three lines are pairwise intersections of three planes), they form a joint. Thus the total number of joints is M(M-1)(M-2)/6. Notice that in this example the number of joints is roughly the number of lines to the 3/2-th power. So J(N) is at least N^(3/2) up to some constant factor.

Upper Bound
Guth and Katz proved in their paper "Algebraic Methods in Discrete Analogs of the Kakeya Problem" that J(N) is O(N^(3/2)). Their proof uses polynomial method and the bound matches the lower bound up to constant factor.

Values of J(N) for small values of N
Main article

The value of J(N) has been found for each N < 11 on the CrowdMath message board.