Rado Theorem Research Articles

Stability of intersecting families

Stability of intersecting families

Erdős–Ko–Rado theorem in Peisert-type graphs

AbstractThe celebrated Erdős–Ko–Rado (EKR) theorem for Paley graphs of square order states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended this result to Peisert graphs and other Cayley graphs which are Peisert-type graphs with nice algebraic properties on the connection set. On the other hand, there are Peisert-type graphs for which the EKR theorem fails to hold. In this article, we show that the EKR theorem of Paley graphs extends to almost all pseudo-Paley graphs of Peisert-type. Furthermore, we establish the stability results of the same flavor.

Vector sum-intersection theorems

Vector sum-intersection theorems

A Kruskal–Katona-type theorem for graphs: q-Kneser graphs

The “Kruskal-Katona-type problem for a graph G” concerned here is to describe subsets of vertices of G that have minimum number of neighborhoods with respect to their sizes. In this paper, we establish a Kruskal-Katona-type theorem for the q-Kneser graphs, whose vertex set consists of all k-dimensional subspaces of an n-dimensional linear space over a q-element field, two subspaces are adjacent if they have the trivial intersection. It includes as a special case the Erdős–Ko–Rado theorem for intersecting families in finite vector spaces and yields a short proof of the Hilton-Milner theorem for nontrivial intersecting families in finite vector spaces.

Erdős–Ko–Rado theorem for vector spaces over residue class rings

Let [Formula: see text] be its decomposition into a product of powers of distinct primes, and [Formula: see text] be the residue class ring modulo [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-dimensional row vector space over [Formula: see text]. A generalized Grassmann graph over [Formula: see text], denoted by [Formula: see text] ([Formula: see text] for short), has all [Formula: see text]-subspaces of [Formula: see text] as its vertices, and two distinct vertices are adjacent if their intersection is of dimension [Formula: see text], where [Formula: see text]. In this paper, we determine the clique number and geometric structures of maximum cliques of [Formula: see text]. As a result, we obtain the Erdős–Ko–Rado theorem for [Formula: see text].

Embedding Dimensions of Simplicial Complexes on Few Vertices

We provide a simple characterization of simplicial complexes on few vertices that embed into the d-sphere. Namely, a simplicial complex on d+3 vertices embeds into the d-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.

Intersection theorems for finite general linear groups

AbstractA subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$ .

Erdős–Ko–Rado and Hilton–Milner theorems for two-forms

Erdős–Ko–Rado and Hilton–Milner theorems for two-forms

A STUDY OF SOME PROPERTIES FOR CELLULAR-LINDELOF TOPOLOGICAL SPACES

Based on Bella and Spadaro (2017), a topological space T is called cellular-Lindelof if for every family F of pairwise disjoint nonempty open subsets of T, there exists a Lindelof subspace L subset T such that F intersection L ≠ empty set for every F∈F.
 The aim of this paper is to investigate the properties of cellular Lindelof spaces, its relation with other spaces and find cardinal inequality for cellular-Lindelof spaces based on the Erdos and Rado theorem. The concept of cellular-Lindelof was utilized to show the properties of cellular-Lindelof spaces and its relation with other spaces. We obtain few examples of topological spaces which are not cellular-Lindelof. Erdos and Rado theorem is a theorem based on intersecting set families. Cardinal inequality that was found is based on lemma from Erdos and Rado theorem and information obtained from some journal papers. The study of cellular-Lindelof spaces is the extension study of Lindelof spaces and this is important as it can become a reference material for the future researchers. The central object of the study of topological dynamics is a topological dynamical system, where topological spaces needed. Thus, the study of cellular-Lindelof spaces is also important to the field of dynamical system.

An asymmetric random Rado theorem: 1-statement

A classical result by Rado characterises the so-called partition-regular matrices A , i.e. those matrices A for which any finite colouring of the positive integers yields a monochromatic solution to the equation A x = 0 . We study the asymmetric random Rado problem for the (binomial) random set [ n ] p in which one seeks to determine the threshold for the property that any r -colouring, r ≥ 2 , of the random set has a colour i ∈ [ r ] admitting a solution for the matrical equation A i x = 0 , where A 1 , … , A r are predetermined partition-regular matrices pre-assigned to the colours involved. We prove a 1-statement for the asymmetric random Rado property. In the symmetric setting our result retrieves the 1-statement of the symmetric random Rado theorem established in a combination of results by Rödl and Ruciński [34] and by Friedgut, Rödl and Schacht [11] . We conjecture that our 1-statement in fact unveils the threshold for the asymmetric random Rado property, yielding a counterpart to the so-called Kohayakawa-Kreuter conjecture concerning the threshold for the asymmetric random Ramsey problem in graphs. We deduce the aforementioned 1-statement for the asymmetric random Rado property after establishing a broader result generalising the main theorem of Friedgut, Rödl and Schacht from [11] . The latter then serves as a combinatorial framework through which 1-statements for Ramsey-type problems in random sets and (hyper)graphs alike can be established in the asymmetric setting following a relatively short combinatorial examination of certain hypergraphs. To establish this framework we utilise a recent approach put forth by Mousset, Nenadov and Samotij [26] for the Kohayakawa-Kreuter conjecture.

Stability of Erdős–Ko–Rado theorems in circle geometries

AbstractCircle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three‐dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family in one of the known finite circle geometries of order , with , must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.

A generalization of Alon–Aydinian–Huang theorem

Let S n , k m be the collection of sets of real numbers of size n , in which every subset of size larger than k has a sum less than m , where n ≥ k + 1 , and m is some real number. Denote by a n , k m the maximum number of nonempty subsets of a set in S n , k m with a sum at least m . In particular, when m = 0 , Alon, Aydinian, Huang ((2014) [1] ) proved that a n , k 0 = ∑ i = 0 k − 1 ( n − 1 i ) , where two technical proofs, based on a weighted version of Hall's theorem and an extension of the nonuniform Erdős–Ko–Rado theorem, were presented. In this note, we extend their elegant result from m = 0 to any real number m , and show that a n , k m = { ∑ i = 0 k − 1 ( n − 1 i ) if m ≥ 0 ∑ i = 1 k ( n i ) if m < 0 . Our proof is obtained by exploring the recurrence relation and initial conditions of a n , k m .

Sharp threshold for the Erdős–Ko–Rado theorem

AbstractFor positive integers and with , the Kneser graph is the graph with vertex set consisting of all ‐sets of , where two ‐sets are adjacent exactly when they are disjoint. The independent sets of are ‐uniform intersecting families, and hence the maximum size independent sets are given by the Erdős–Ko–Rado Theorem. Let be a random spanning subgraph of where each edge is included independently with probability . Bollobás, Narayanan, and Raigorodskii asked for what does have the same independence number as with high probability. For , we prove a hitting time result, which gives a sharp threshold for this problem at . Additionally, completing work of Das and Tran and work of Devlin and Kahn, we determine a sharp threshold function for all .

An algebraic groups perspective on Erdős–Ko–Rado

We give a proof of the Erdős–Ko–Rado Theorem using the Borel Fixed-Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erdős–Ko–Rado Theorem and (generalizations of) the Gerstenhaber Theorem on spaces of nilpotent matrices.

The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

An asymmetric random Rado theorem for single equations: The 0‐statement

AbstractA famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring of the positive integers gives rise to a monochromatic solution to the equation . Aigner‐Horev and Person recently stated a conjecture on the probability threshold for the binomial random set having the asymmetric random Rado property: given partition regular matrices (for a fixed ), however one r‐colors , there is always a color such that there is an i‐colored solution to . This generalizes the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner‐Horev and Person proved the 1‐statement of their asymmetric conjecture. In this paper, we resolve the 0‐statement in the case where the correspond to single linear equations. Additionally we close a gap in the original proof of the 0‐statement of the (symmetric) random Rado theorem.

Beyond the Erdős Matching Conjecture

A family F ⊂ [ n ] k is U ( s , q ) of for any F 1 , … , F s ∈ F we have | F 1 ∪ ⋯ ∪ F s | ≤ q . This notion generalizes the property of a family to be t -intersecting and to have matching number smaller than s . In this paper, we find the maximum | F | for F that are U ( s , q ) , provided n > C ( s , q ) k with moderate C ( s , q ) . In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization of the Erdős–Ko–Rado theorem, which states that for n > s 2 k the largest family F ⊂ [ n ] k with property U ( s , s ( k − 1 ) + 1 ) is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in [ n ] k is U ( s , s ( k − 1 ) + 1 ) .) We investigate the case k = 3 more thoroughly, showing that, unlike in the case of the Erdős Matching Conjecture, in general there may be 3 extremal families.

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

Rainbow Odd Cycles

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 16 November 2020Accepted: 13 June 2021Published online: 04 October 2021Keywordsrainbow cycle, odd cycle, cactus graph, Rado's theorem for matroidsAMS Subject Headings05C38, 05C70, 05B35Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

On the size of shadow-added intersecting families

Let n ≥ 2 k − 1 > 1 , [ n ] = { 1 , 2 , … , n } . For a family F of k -subsets of [ n ] let ∂ F be the immediate shadow (cf. Definition 1.1 ) of F . Suppose that | F ∩ F ′ | ≥ 2 for all F , F ′ ∈ F . We conjecture that | F | + | ∂ F | ≤ 3 n − 2 k − 2 + n − 2 k − 3 and prove it for n = 2 k − 1 , n ≥ 3 ( k − 1 ) and also for k ≤ 10 . This problem is somewhat unusual but we exhibit deep connections to the Erdős–Ko–Rado Theorem and to the Erdős Matching Conjecture. Some related problems are also considered.