Related papers: Sums and products along sparse graphs
In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this circle of ideas, we resolve a conjecture of Bremner on…
In 1959, Erd\H{o}s and Szekeres posed a series of problems concerning the size of polynomials of the form $$ P_n(z) = \prod_{j=1}^n (1 - z^{s_j}), $$ where $s_1, \dots, s_n$ are positive integers. Of particular interest is the quantity $$…
Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di [Discrete Math 13 (1975), 97--107] investigated a tripartite generalization of the Zarankiewicz problem: what minimum degree forces a tripartite graph with $n$ vertices in each part to contain an…
The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has…
In this paper we continue the study of a natural generalization of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by [Thomassen '89] and a linear-time fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive…
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…
The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…
A set cover of a hypergraph $H$ is a set of vertices intersecting every hyperedge. In the minimum sum set cover problem, vertices are selected one by one; each edge pays the position of the first vertex that hits it, and the objective is to…
An old problem raised independently by Jacobson and Sch\"onheim asks to determine the maximum $s$ for which every graph with $m$ edges contains a pair of edge-disjoint isomorphic subgraphs with $s$ edges. In this paper we determine this…
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…
A set of cycles is called independent if no two of them have a common vertex. Let $S_{n, 2k-1}$ be the complete split graph, which is the join of a clique of size $2k-1$ with an independent set of size $n-2k+1$. In 1962, Erd\H{o}s and…
Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \[ \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq…
Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an…
We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that spectra of…
New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising…
An $(n,s,q)$-graph is an $n$-vertex multigraph in which every $s$-set of vertices spans at most $q$ edges. Erd\H{o}s initiated the study of maximum number of edges of $(n,s,q)$-graphs, and the extremal problem on multigraphs has been…
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut…
We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each…