Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a $B_k^+$-set but the converse is not true in general. Determining the largest size of a $B_k$-set in the interval $\{1, 2, \dots, N \} \subset \integers$ or in the cyclic group $\integers_N$ is a well studied problem. In this paper we investigate the corresponding problem for $B_k^+$-sets. We prove non-trivial upper bounds on the maximum size of a $B_k^+$-set contained in the interval $\{1, 2, \dots, N \}$. For odd $k \geq 3$, we construct $B_k^+$-sets that have more elements than the $B_k$-sets constructed by Bose and Chowla. We prove a $B_3^+$-set $A \subset \integers_N$ has at most $(1 + o(1))(8N)^{1/3}$ elements. Finally we obtain new upper bounds on the maximum size of a $B_k^*$-set $A \subset \{1,2, \dots, N \}$, a problem first investigated by Ruzsa.
We prove that if $A\subseteq \{ 1,2,\dots, N \}$ does not contain any solution to the equation $x_1+\dots+x_k=y_1+\dots+y_k$ with distinct $x_1,\dots,x_k,y_1,\dots,y_k\in A$, then $|A|\ll {k^{3/2}}N^{1/k}.$
We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemerédi.
We have the Fisher type inequality and the linear programming bound as upper bounds for the cardinalities of $s$-distance sets on $S^{d-1}$. In this paper, we give a new upper bound for the cardinalities of $s$-distance sets on $S^{d-1}$ for any $s$. This upper bound improves the Fisher typer inequality and is useful for $s$-distance sets which are not applicable to the linear programming bound.
For $g \geq 2$ and $h \geq 3$, we give small improvements on the maximum size of a $B_h[g]$-set contained in the interval $\{1,2, \dots , N \}$. In particular, we show that a $B_3[g]$-set in $\{1,2, \dots , N \}$ has at most $(14.3 g N)^{1/3}$ elements. The previously best known bound was $(16 gN)^{1/3}$ proved by Cilleruelo, Ruzsa, and Trujillo. We also introduce a related optimization problem that may be of independent interest.
We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance graphs.
We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ is at most $π(n)+π(n/2)+n^{2/3}(\log n )^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $π(n)+π(n/2)+cn^{2/3}\log n/\log\log n$.
The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin's set. Th…
In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.
We present a strengthening of the lemma on the lower bound of the slice rank by Tao (2016) motivated by the Croot-Lev-Pach-Ellenberg-Gijswijt bound on cap sets (2017, 2017). The Croot-Lev-Pach-Ellenberg-Gijswijt method and the lemma of Tao are based on the fact that the rank of a diagonal matrix is equal to the number of non-zero diagonal entries. Our lemma is based on the rank of upper-triangular matrices. This stronger lemma allows us to prove the following extension of the Ellenberg-Gijswijt result (2017). A tricolored ordered sum-free set in $\mathbb F_p^n$ is a collection $\{(a_i,b_i,c_i)…
A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of…
We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains.
The following Proposition is a positive answer to a question about cancellations between permutations that arises in a model problem in the many body theory of Fermions. It concerns the mathematically rigorous implementation of the Pauli exclusion principle.
We show that any subset of $\mathbb{Z}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}<1$. In particular, we find an upper bound of $O(2.84^n)$ on the maximum size of an affine cap in $GF(3)^n$.
A finite set X in the Euclidean space is called an s-inner product set if the set of the usual inner products of any two distinct points in X has size s. First, we give a special upper bound for the cardinality of an s-inner product set on concentric spheres. The upper bound coincides with the known lower bound for the size of a Euclidean 2s-design. Secondly, we prove the non-existence of 2- or 3-inner product sets on two concentric spheres attaining the upper bound for any d>1. The efficient property needed to prove the upper bound for an s-inner product set gives the new concept, inside s-in…
We prove an extension of Bourgain's theorem on pinned distances in measurable subset of $\mathbb{R}^2$ of positive upper density, namely Theorem $1^\prime$ in [Bourgain, 1986], to pinned non-degenerate $k$-dimensional simplices in measurable subset of $\mathbb{R}^{d}$ of positive upper density whenever $d\geq k+2$ and $k$ is any positive integer.
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 Croot, Lev, and Pach in their breakthrough work on arithmetic-progression-free sets, we prove that the size of any tri-colored sum-free set in $\mathbb{F}_2^n$ is bounded above by $6 {n \choose \lfloor n/3 \rfloor}$. This upper bound is tight, up to a factor subexponential in $n$: there exist tri-colored sum-free sets in $\mathbb{F}_2^n$ of size greater than ${…
In a recent paper Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester's Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mbox{$\cal A$}\subseteq {\mathbb R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-…
We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed $n$-dimensional rectangular box, the same vertices extended with $n$ points completing three-term arithmetic progressions, and the same vertices extended with $n$ points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.
The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{α, -α\}$, $α\in[0,1)$, are called equiangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $\ma…
We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.
We answer a question of Hegyvári and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group, and it does not require all addends in the triple products we consider to have positive (left) upper Banach densities; one of the addends is allowed to only have positive upper asymptotic density with respect to a (possibly very sparse) ergodic sequence.
We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts) partitions of an integer n.
A subset $A$ of the integers is a $B_k[g]$ set if the number of multisets from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote the maximum size of a $B_k[g]$ set in $\{1,\dots, n\}$. In this paper we improve the best-known upper bounds on $F_{k,g}(n)$ for $g>1$ and $k$ large. When $g=1$ we match the best upper bound of Green with an improved error term. Additionally, we give a lower bound on $F_{k,g}(n)$ that matches a construction of Lindström while removing one of the hypotheses.
We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss showing a similar property for m=k=2.
We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small …
A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distinct distances between two distinct points in $X$ and a subset $X$ is called a locally $k$-distance set if for any point $x$ in $X$, there are at most $k$ distinct distances between $x$ and other points in $X$. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of $k$-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally $k$-distance sets on a sphere. In the first part of this paper, we prove that if $X$ is…
In this paper we study the the average order of dominating sets in a graph, $\operatorname{avd}(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown (2021) conjectured that for all graphs $G$ of order $n$ without isolated vertices, $\operatorname{avd}(G) \leq 2n/3$. Recently, Erey (2021) proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $\operatorname{avd}(G) = 2n/3$. We also use our bounds to prove the average version of Vizing's Conjecture.
The aim of this paper is to obtain new inequalities for a large family of generalizations of the Wiener Index and to characterize the set of extremal graphs with respect to them. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs.
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. We also apply these bounds to the adjacency spectral radius and signless Laplacian spectral radius of a uniform hypergraph.
In his 1992 Ph.D. thesis Chang identified an efficient way to dominate $m \times n$ grid graphs and conjectured that his construction gives the most efficient dominating sets for relatively large grids. In 2011 Gonçalves, Pinlou, Rao, and Thomassé proved Chang's conjecture, establishing a closed formula for the domination number of a grid. In March 2013 Fata, Smith and Sundaram established upper bounds for the $k$-distance domination numbers of grid graphs by generalizing Chang's construction of dominating sets to $k$-distance dominating sets. In this paper we improve the upper bounds establis…
The size of a largest independent set of vertices in a given graph $G$ is denoted by $α(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether $α(G) \geq K.$ An upper bound for the independence number $α(G)$ of a given graph $G$ with $n$ vertices and $m $ edges is given by $α(G) \leq p:=\lfloor\frac{1}{2} + \sqrt{\frac{1}{4} + n^2 - n - 2m}\rfloor.$ In this paper we will consider maximum independent sets near this upper bound. Our main result is the following: There exists an algorithm with time complexity $O(n…
We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$ and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case of $s=0$. We characterize the minimizers and provide examples of maximizers, for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of indepen…
In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonngeative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve bounds on Perron values of rooted trees in combinatorial settings, in order to find characteristic sets of trees. The other is to generate log-concave and log-convex sequences through the monotone sequences.
We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan, Bush, Croot, and Pryby. The upper bounds are based on a version of the asymmetric Balog-Szemeredi-Gowers theorem for group actions combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and amenability (or expanding families of graphs in the finite field case). As an application of…
Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If true, this bound would be tight, being achieved by the disjoint union of $N/2d$ copies of $K_{d,d}$. Kahn established the bound for bipartite $G$, and later gave an argument that established $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{2}{d}\right)} $$ for $G$ not necessarily bipartite. In this note, we improve this to $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{1+o(1…
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.
Let $q,d\geq 2$ be integers. Define $$ J(q,d):=\frac 1q \Big( \min_{0<x<1} \frac{1-x^q}{1-x} x^{-\frac{q-1}{d}}\Big). $$ Let $\mbox{$\cal G$}\subseteq {\mathbb R}^n$ be an arbitrary subset. We denote by $d(\mbox{$\cal G$})$ the set of (non-zero) distances among points of $\mbox{$\cal G$}$: $$ d(\mbox{$\cal G$}):=\{d( p_1, p_2):~ p_1, p_2\in \mbox{$\cal G$}, p_1\ne p_2\}. $$ Our main result is a new upper bound for the size of $s$-distance sets in boxes. More concretely, let $A_i\subseteq \mathbb R$, $|A_i|=q\geq 2$ be subsets for each $1\leq i\leq n$. Consider the box $\mbox{$\cal B$}:=\prod_{…
The aim of this paper is to obtain new inequalities for a large family of topological indices restricted to unicyclic graphs and to characterize the set of extremal unicyclic graphs with respect to them. This family includes variable first Zagreb, variable sum exdeg, multiplicative second Zagreb and Narumi-Katayama indices. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs, fixing or not the maximum degree or the number of pendant vertices.
We establish that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided $d\geq4$. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if $Δ_{k_1}$ and $Δ_{k_2}$ are two fixed non-degenerate simplices of $k_1+1$ and $k_2+1$ points respectively, then any subset of $\mathbb{R}^d$ of positive upper Banach density with $d\geq k_1+k_2+6$ will necessarily contain an isometric copy o…
