Related papers: An upper bound on tricolored ordered sum-free sets

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by…

In Ellenberg and Gijswijt's groundbreaking work, the authors show that a subset of $\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-\epsilon)^n)$), and…

Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\bf a}_i, {\bf b}_i, {\bf c}_i)$ in $G^n$ such that ${\bf a}_i+{\bf b}_j+{\bf c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be…

Inspired by the Croot-Lev-Pach breakthrough, Jordan Ellenberg and Dion Gijswijt have recently amazed the combinatorial world by proving that the largest size of a subset of $F_3^n$ with no 3-term arithmetic progressions is exponentially…

In 2016, Ellenberg and Gijswijt established a new upper bound on the size of subsets of $\mathbb{F}^n_q$ with no three-term arithmetic progression. This problem has received much mathematical attention, particularly in the case $q = 3$,…

Recently, Croot, Lev, and Pach (Ann. of Math., 185:331--337, 2017.) and Ellenberg and Gijswijt (Ann. of Math., 185:339--443, 2017.) developed a new polynomial method and used it to prove upper bounds for three-term arithmetic progression…

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $\omega$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans…

The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem…

Let $S$ and $T$ be subsets of $\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of…

Croot, Lev and Pach used a new polynomial technique to give a new exponential upper bound for the size of three-term progression-free subsets in the groups $(\mathbb Z _4)^n$. The main tool in proving their striking result is a simple lemma…

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…

We show that the slice rank of the direct sum of two tensors is equal to the sum of their slice ranks. The upper bound is trivial, but the lower bound needs more than a one-line proof, for reasons we explain. This result generalizes the…

In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd…

Ellenberg and Gijswijt gave the best known asymptotic upper bound for the cardinality of subsets of $\mathbb F_q^n$ without 3-term arithmetic progressions. We improve this bound by a factor $\sqrt{n}$. In the case $q=3$, we also obtain more…

By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $\Delta$ has chromatic number at most $(C+o(1))\Delta/\log \Delta$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one…

We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve…

Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$.…

We prove that the stack-number of the strong product of three $n$-vertex paths is $\Theta(n^{1/3})$. The best previously known upper bound was $O(n)$. No non-trivial lower bound was known. This is the first explicit example of a graph…

In this paper we rectify two previous results found in the literature. Our work leads to a new upper bound for the largest sum-free subset of $[1,n]$ with lowest value in $\left [\frac{n}{3},\frac{n}{2}\right ]$, and the identification of…

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever…