Related papers: On sums and products along the edges, II
In their seminal paper Erd\H{o}s and Szemer\'edi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show…
In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any $n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They…
This is a short survey article written for the Erd\H{o}s centennial conference in Budapest in 2013. The main two topics covered are Szemer\'edi's theorem and its ramifications, and the Erd\H{o}s discrepancy problem. There is an emphasis on…
In this short note, we establish an edge-isoperimetric inequality for arbitrary product graphs. Our inequality is sharp for subsets of many different sizes in every product graph. In particular, it implies that the $2^d$-element sets with…
The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In this paper, we study the bounds on the sum and product of the independence numbers of graphs and their line…
In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…
A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, recall their theorem, which gives bounds on the sum and…
In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.
We introduce a new method for decomposing the edge set of a graph, and use it to replace the Regularity lemma of Szemer\'edi in some graph embedding problems. An algorithmic version is also given.
We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field…
We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erd\H{o}s. We also mention some related problems.
A multigraph G is said to be an (s,q)-graph if every s-set of vertices in G supports at most q edges (counting multiplicities). In this paper we consider the maximal sum and product of edge multiplicities in an (s,q)-graph on n vertices.…
We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erd\H{o}s-Stone theorem. We also provide means…
Erd\H{o}s proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.
A new product construction of graphs and digraphs, based on the standard box product of graphs and called the separated box product, is presented, and several of its properties are discussed. Questions about the symmetries of the product…
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the…
We characterize strongly edge regular product graphs and find the edge-balanced index sets of complete bipartite graphs without a perfect matching, the direct product $K_n\times K_2$. We also prove a lemma that is helpful to determine the…
We present some new sharp constructions for the Szemer\'{e}di-Trotter theorem. These constructions generalize previous work of Erd\H{o}s, Elekes, Sheffer and Silier, Guth and Silier, and the author. In the past, arguments showing the…
We resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If $G$ is a connected $d$-regular graph with $n$ vertices, the graph square of $G$ is not complete, and $G$ is not a member of two narrow…
We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or…