A lower bound for the kk-multicolored sum-free problem in Zmn\mathbb{Z}^n_m

In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$, where $k\geq 3$ and $m\geq 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$. This generalizes a result of Kleinberg-Sawin-Speyer for the case $k=3$ and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg-Sawin-Speyer. Both of these generalizations require several key new ideas

The asymptotic induced matching number of hypergraphs: balanced binary strings

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science. Phrased in the language of tensors, as a direct consequence of our result, we determine the asymptotic subrank of any tensor with support given by the aforementioned hypergraphs. In the context of quantum information theory, our result amounts to an asymptotically optimal $k$-party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement

Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur

The growth rate of tri-colored sum-free sets

The growth rate of tri-colored sum-free sets, Discrete Analysis 2018:12, 10 pp. This paper contributes to the remarkable collection of results that followed in the wake of the 2016 breakthrough by Ellenberg and Gijswijt on the cap set problem, which asks for the maximal size of a 3-term progression-free subset of $\mathbb{F}_3^n$. The polynomial method of Ellenberg and Gijswijt, who followed the lead of Croot, Lev, and Pach after whom the method is now named, showed for the first time that the size of such a set is bounded by a polynomial in the size of the ambient space. More specifically, they showed that a cap set in $\mathbb{F}_3^n$ can be of size at most $(2.756)^n$. This paper considers a variant of the cap set problem, namely the question of how large a tri-coloured sum-free subset of $\mathbb{F}_3^n$ can be. By a tri-coloured sum-free subset we mean a collection of triples $(a_i,b_i,c_i)$ in $(\mathbb{F}_3^n)^3$ such that $a_i+b_j+c_k=0$ if and only if $i=j=k$. Note that a cap set $A\subseteq \mathbb{F}_3^n$ gives rise to a tri-coloured sum-free set, namely the collection of triples $\{(a,a,a): a\in A\}$. It is therefore not entirely surprising that the Croot-Lev-Pach polynomial method can be used to show that if $q$ is a prime power, then a tri-coloured sum-free set in $\mathbb{F}_q^n$ can have size at most $3\theta^n$, where $\theta$ is the solution to an explicit optimisation problem. This is one of the results appearing in the paper ["On cap sets and the group-theoretic approach to matrix multiplication"](http://discreteanalysisjournal.com/article/1245), by Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans, which Discrete Analysis published earlier this year. In the present paper the authors show this bound to be tight to within a subexponential factor. The lower bound is based on a construction of an $S_3$-symmetric probability distribution on $\{(a,b,c)\in\mathbb{Z}_{\geq 0}^3:a+b+c=n\}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,…,n\}$ with mean $n/3$. In answer to a conjecture which was posed in the original preprint version of this article, the existence of such a distribution was established by Pebody, whose [article](http://discreteanalysisjournal.com/article/3733-proof-of-a-conjecture-of-kleinberg-sawin-speyer) is being published in Discrete Analysis alongside the present paper. The question of whether the Croot-Lev-Pach polynomial method also yields a tight bound for the cap set problem remains open