In this paper, we prove that x choose (d-1) lines in a d-dimensional space determine at most x choose d  joints. This is tight for integral x, and we also determine all the configurations meeting this upper bound. 

Inspired by the joints problem, we consider several graph-theoretic problems where the number of edges is given and we are asked to determine the maximum number of vertex subsets with the desired induced structure. We focus on two particular instances of such settings: determining the maximum number of rainbow triangles given the numbers of red, green and blue edges, and the Kruskal--Katona theorem. We use the entropy method to show some strong bounds in both cases, and list some interesting open problems of the same flavor.

In the joints problem, we may consider a variant where lines are replaced with two-dimensional flats or even some other varieties in general. In this paper, we resolve the joints problem for varieties up to a multiplicative constant factor, and our result also generalizes all known variants of the joints problem.

See here for an online talk given by Yufei.

In this paper, we resolve the joints problem (for lines) up to an o(1)-factor. The main ingredient is a new vanishing lemma designed for the joints problem leading to an inequality with parameters that we may choose.

A sock ordering  is a sequence of socks with some colors, and a sock ordering is foot-sortable if it can be rearranged using a stack so that socks of the same color form a contiguous block. In this paper, I show how foot-sortability can be determined in O(N log N) time, and also classify all the minimal sock orderings with no colors appearing more than twice that are not foot-sortable.