On the Ramsey Number of Sparse 3-Graphs
AbstractWe consider a hypergraph generalization of a conjecture of Burr and Erdős concerning the Ramsey number of graphs with bounded degree. It was shown by Chvátal, Rödl, Trotter, and Szemerédi [The...
View ArticleClik here to view.
Cliques in Steiner systems
AbstractA partial Steiner (k,l)-system is a k-uniform hypergraph with the property that every l-element subset of V is contained in at most one edge of . In this paper we show that for given k,l and t...
View ArticleGeneralizations of the removal lemma
AbstractRuzsa and Szemerédi established the triangle removal lemma by proving that: For every η>0 there exists c>0 so that every sufficiently large graph on n vertices, which contains at most...
View ArticleRegularity Lemmas for Graphs
AbstractSzemerédi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some...
View ArticleColoring Uniform Hypergraphs with Small Edge Degrees
AbstractLet k be a positive integer and n=⌊log2k⌋. We prove that there is an ε = ε(k) > 0 such that for sufficiently large r, every r-uniform hypergraph with maximum edge degree at most $$...
View ArticleOn the Function of Erdős and Rogers
AbstractIn 1930, Frank Ramsey published a seminal paper “On a problem of formal logic”[13] beginning a new area of research known today as Ramsey theory (for a comprehensive introduction to Ramsey...
View ArticleA Deterministic Algorithm for the Frieze-Kannan Regularity Lemma
AbstractThe Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial...
View ArticleOn K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers
AbstractExtending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s< t let fs,t(n) = min{max{|S|: S⊆ V (H) and H[S] contains no...
View ArticleHereditary quasirandom properties of hypergraphs
AbstractThomason and Chung, Graham and Wilson were the first to systematically investigate properties of quasirandom graphs. They have stated several quite disparate graph properties — such as having...
View ArticleAn Improved Upper Bound on the Density of Universal Random Graphs
AbstractWe give a polynomial time randomized algorithm that, on receiving as input a pair (H,G) of n-vertex graphs, searches for an embedding of H into G. If H has bounded maximum degree and G is...
View ArticleExtremal Results in Random Graphs
AbstractAccording to Paul Erdős au][Some notes on Turán’s mathematical work, J. Approx. Theory 29 (1980), page 4]_it was Paul Turán who “created the area of extremal problems in graph theory”. However,...
View ArticleArrangeability and Clique Subdivisions
SummaryLet k be an integer. A graph G is k-arrangeable (concept introduced by Chen and Schelp) if the vertices of G can be numbered v1, v2, …, vn in such a way that for every integer i with 1 ≤ i≤ n,...
View ArticleThe Complexity of Proving That a Graph Is Ramsey
AbstractWe say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c logn. We define a CNF formula which expresses this property for a graph G....
View ArticleThe complexity of proving that a graph is Ramsey
AbstractWe say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph...
View ArticleAn exponential-type upper bound for Folkman numbers
AbstractFor given integers k and r, the Folkman number f(k;r) is the smallest number of vertices in a graph G which contains no clique on k+1 vertices, yet for every partition of its edges into r...
View ArticleDiscrepancy and eigenvalues of Cayley graphs
AbstractWe consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are...
View ArticleStatistics of orderings
AbstractIn this paper, we study a Ramsey type problems dealing with the number of ordered subgraphs present in an arbitrary ordering of a larger graph. Our first result implies that for every vertex...
View ArticleOn the Hamiltonicity of Triple Systems with High Minimum Degree
AbstractWe show that every 3-uniform hypergraph with minimum vertex degree at least 0.8 \(\left(\begin{array}{c}n-1\\2\end{array}\right)\) contains a tight Hamiltonian cycle.
View ArticleRamsey Partial Orders from Acyclic Graphs
AbstractWe prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that the class of acyclic graphs has the Ramsey property and uses the partite...
View ArticleA Note on Induced Ramsey Numbers
AbstractThe induced Ramsey number rind(F) of a k-uniform hypergraph F is the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that every two-coloring of the...
View Article
