# A gap in the slice rank of $k$-tensors

## TL;DR

This paper establishes fundamental limits on the slice rank of k-tensors, showing it is either 1 or at least a specific bound, indicating the method's limitations in certain combinatorial problems.

## Contribution

It proves a lower bound on the asymptotic slice rank of any k-tensor, revealing inherent limitations of the slice-rank method in combinatorial bounds.

## Key findings

- Slice rank of k-tensors is either 1 or at least a specific bound.
- The result indicates limitations of the slice-rank method for certain problems.
- Provides evidence that the method cannot always produce non-trivial bounds.

## Abstract

The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been introduced in different contexts but little is known about the limitations of the method. In this paper, building upon a method presented by Tao and Sawin, it is proved that the asymptotic slice rank of any $k$-tensor in any field is either $1$ or at least $k/(k-1)^{(k-1)/k}$. This provides evidence that straight-forward application of the method cannot give useful results in certain problems for which non-trivial exponential bounds are already known. An example, actually a motivation for starting this work, is the problem of bounding the size of trifferent sets of sequences, which constitutes a long-standing open problem in information theory and in theoretical computer science.