Triangular tensors and set-intersection problems

TL;DR

This paper introduces triangular tensors, generalizes the slice-rank lemma, and applies this framework to provide new proofs and improvements for set-intersection theorems and related combinatorial problems.

Contribution

It defines triangular tensors, extends the slice-rank lemma to them, and applies this to simplify proofs and improve bounds in combinatorics.

Findings

New proof of Frankl-Wilson theorem

Improved bounds for Snevily's theorem

Generalized bounds for reverse odd-town problem

Abstract

In the past few years, the slice-rank lemma of Tao has been applied successfully to many problems in extremal combinatorics. In this paper, first, we define a new notion of triangular tensors which generalizes that of triangular matrices (2-tensors), and prove a lemma similar to the slice-rank lemma for them. Then, applying the slice-rank framework with triangular matrices, we give new and shorter proofs for some well-known theorems on set-intersections like Frankl-Wilson and Snevily with modular constraints, and some of the more recent set-intersection results. We also improve Snevily with modular constraints in some special cases. Finally, using Snevily's theorem with some combinatorial lemmas, we give new bounds on some generalizations of the reverse odd-town problem.