Fox–Himwich–Zhou TAS-tree conjecture

An undirected tree is a connected graph with no cycles; an orientation assigns a direction to each of its edges. A directed graph has the tournament anti-Sidorenko property (TAS) when its homomorphism density into every tournament satisfies the anti-Sidorenko inequality.

TAS-tree conjecture. For every undirected tree, there exists some orientation that has the tournament anti-Sidorenko property.

This conjecture was posed by Fox, Himwich, Zhou, and the second author. It is known for trees with exactly one vertex of even degree, but remains open in general.