Triangular tensors and set-intersection problems
Omran Ahmadi, Hassan Norouzi
TL;DR
The paper expands the slice-rank toolbox by introducing triangular tensors, including triangular $2$-tensors and general $i$-triangular $k$-tensors, and proves a slice-rank lemma for them on totally ordered sets. This framework is applied to modular set-intersection problems to yield concise, self-contained proofs of classical results such as Frankl–Wilson and Snevily with modular constraints, and to establish new bounds for generalized reverse odd-town problems. Through combinatorial lemmas like the Complement and Trace Lemmas, the authors derive sharp bounds and reductions that extend to higher-order intersection profiles. Overall, the triangular-tensor approach provides a versatile, streamlined method for obtaining bounds in extremal set theory under modular constraints, with potential for further refinements in reverse odd-town type problems and beyond.
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.