Related papers: Erd\H{o}s-Rado Classes

The Erd\H{o}s--Ko--Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes,…

We establish a variety of extensions to the Erdos-Rado Theorem, particularly involving ordinal numbers, and always involving ordinary partition relations. Most of the results can be regarded as consequences of the Ramification Principle,…

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Tur\'an's theorem, the…

In this short note we show that both generalizations of celebrated Erd\H{o}s--Ko--Rado theorem and Hilton--Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from the folklore puzzle about…

There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical…

Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erd\H{o}s-Rado sunflower lemma.…

The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of {1,2,...,n} (n>2k-1) such that every pair of elements in F has a nonempty intersection, then |F| is at most $\binom{n-1}{k-1}$. The theorem also…

This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey property, a strong combinatorial…

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…

Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $\mathbb{Z}_{h}^{n}$ be the $n$-dimensional row vector space over…

The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erd\H{o}s, Chao Ko and Richard Rado proved the (first) `Erd\H{o}s-Ko-Rado theorem' on the maximum possible size of an intersecting family of…

The Erd\H{o}s-Rado canonization theorem generalizes Ramsey's theorem to edge-colorings with an unbounded number of colors, in the sense that for $n = ER(m)$ sufficiently large, any edge-coloring of $E(K_n) \to \mathbb{N}$ will yield some…

Intersecting and cross-intersecting families usually appear in extremal combinatorics in the vein of the Erd{\H o}s--Ko--Rado theorem. On the other hand, P.~Erd{\H o}s and L.~Lov{\'a}sz in the noted paper~\cite{EL} posed problems on…

We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank…

Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…

The main topic of this paper is various "hyperbolic" generalizations of the Edmonds-Rado theorem on the rank of intersection of two matroids. We prove several results in this direction and pose a few questions. We also give generalizations…

We give a proof of the Erd\H{o}s-Ko-Rado Theorem using the Borel Fixed Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erd\H{o}s-Ko-Rado Theorem and (generalizations of) the Gerstenhaber…

We establish some new generalizations of Erd\H{o}s-Mordell inequality by adding weights to its terms. Using these generalizations, we derived strengthened versions of the original Erd\H{o}s-Mordell inequality. We also found two other…

Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory,…

In this paper, we will develop a significantly more general notion of classical Ramsey numbers (extending most other graph-theoretic generalizations) and make some preliminary characterizations of these new Ramsey numbers using simple…