scholarly journals On the structure of random hypergraphs

2018 ◽  
Vol 104 (118) ◽  
pp. 43-51
Author(s):  
Borisa Kuzeljevic
Keyword(s):  
Partial Order ◽  
Linear Order ◽  
Minimal Element ◽  
Maximal Chain ◽  
Order Type ◽  
Compact Set ◽  
Uniform Hypergraph ◽  
Random Hypergraphs

Let Hn be a countable random n-uniform hypergraph for n > 2, and P(Hn) = {f[Hn] : f : Hn ? Hn is an embedding}. We prove that a linear order L is isomorphic to the maximal chain in the partial order ?P(Hn)?{?},?? if and only if L is isomorphic to the order type of a compact set of reals whose minimal element is nonisolated.

A Novel Ranking Procedure Based on Risk Percentile Index and Concentration (rrpic) of Chemicals

2012 ◽  
Vol 610-613 ◽  
pp. 762-765
Author(s):  
Fu Chen ◽  
Shu Shen Liu
Keyword(s):  
Partial Order ◽  
Linear Order ◽  
Taihu Lake ◽  
Chemical Properties ◽  
Ecological Impacts ◽  
Order Model ◽  
Ranking Procedure ◽  
Relative Risks ◽  
Physico Chemical ◽  
Management Plans

Assessments of the relative risks posed by chemicals are needed to assist in the development of management plans that minimize ecological impacts. A procedure scoring and screening chemicals entitled ranking procedure based on risk percentile index and concentration (rrpic) was developed based on inherent physico-chemical properties and toxicity as well as detection concentration. The properties and toxicity were transformed into risk percentile indices (rpi) and the rpi was then used to calculate risk score (rs) by multiplying the detection concentration. Using Hasse diagram, the chemicals having different rss were classified into several rank levels. The averaged rank (rav) of each chemical was calculated by using Local Partial Order Model (LPOM) and the risks of chemicals were arranged in a linear order. The rrpic was employed to scoring and screening 31 chemicals detected in Taihu Lake of China.

scholarly journals The Multiple-Orientability Thresholds for Random Hypergraphs

2015 ◽  
Vol 25 (6) ◽  
pp. 870-908 ◽  
Author(s):  
NIKOLAOS FOUNTOULAKIS ◽  
MEGHA KHOSLA ◽  
KONSTANTINOS PANAGIOTOU
Keyword(s):  
Load Balancing ◽  
Hard Disk ◽  
Maximum Load ◽  
Disk Arrays ◽  
Uniform Hypergraph ◽  
Cuckoo Hashing ◽  
Uniform Hypergraphs ◽  
Random Hypergraphs ◽  
Do So ◽  
Parallel Access

Ak-uniform hypergraphH= (V, E) is called ℓ-orientable if there is an assignment of each edgee∈Eto one of its verticesv∈esuch that no vertex is assigned more than ℓ edges. LetHn,m,kbe a hypergraph, drawn uniformly at random from the set of allk-uniform hypergraphs withnvertices andmedges. In this paper we establish the threshold for the ℓ-orientability ofHn,m,kfor allk⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantityc*k,ℓsuch that with probability 1 −o(1) the graphHn,cn,khas an ℓ-orientation ifc<c*k,ℓ, but fails to do so ifc>c*k,ℓ.Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

scholarly journals The Minimum Number of Edges in Uniform Hypergraphs with Property O

2017 ◽  
Vol 27 (4) ◽  
pp. 531-538 ◽  
Author(s):  
DWIGHT DUFFUS ◽  
BILL KAY ◽  
VOJTĚCH RÖDL
Keyword(s):  
Linear Order ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Minimum Number ◽  
Vertex Set

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

Betweenness relations and cycle-free partial orders

1996 ◽  
Vol 119 (4) ◽  
pp. 631-643 ◽  
Author(s):  
J. K. Truss
Keyword(s):  
Partial Order ◽  
Linear Order ◽  
Partial Orders ◽  
Partial Ordering ◽  
Macneille Completion ◽  
Alternative Definition ◽  
Betweenness Relations

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

Bad models in nice neighborhoods

1986 ◽  
Vol 51 (4) ◽  
pp. 1043-1055 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Linear Order ◽  
Homogeneous Model ◽  
Order Type ◽  
Saturated Model ◽  
Countable Number ◽  
Decidable Theory ◽  
Type Spectrum ◽  
Image Position ◽  
Countable Models

This paper contains an example of a decidable theory which has1) only a countable number of countable models (up to isomorphism);2) a decidable saturated model; and3) a countable homogeneous model that is not decidable.By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such thatis a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

scholarly journals Threshold and Hitting Time for High-Order Connectedness in Random Hypergraphs

10.37236/5064 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Oliver Cooley ◽  
Mihyun Kang ◽  
Christoph Koch
Keyword(s):  
High Probability ◽  
Hitting Time ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Time Result ◽  
Edge Probability ◽  
The Moment ◽  
Definition Of ◽  
Random Hypergraph ◽  
Random Hypergraphs

We consider the following definition of connectedness in $k$-uniform hypergraphs: two $j$-sets (sets of $j$ vertices) are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. The hypergraph is $j$-connected if all $j$-sets are pairwise $j$-connected. We determine the threshold at which the random $k$-uniform hypergraph with edge probability $p$ becomes $j$-connected with high probability. We also deduce a hitting time result for the random hypergraph process – the hypergraph becomes $j$-connected at exactly the moment when the last isolated $j$-set disappears. This generalises the classical hitting time result of Bollobás and Thomason for graphs.

scholarly journals Saturation Number of Berge Stars in Random Hypergraphs

10.37236/9302 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Lele Liu ◽  
Changxiang He ◽  
Liying Kang
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Minimum Number ◽  
Saturation Number ◽  
Random Hypergraphs

Let $G$ be a graph. We say an $r$-uniform hypergraph $H$ is a Berge-$G$ if there exists a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each $e\in E(G)$. Given a family of $r$-uniform hypergraphs $\mathcal{F}$ and an $r$-uniform hypergraph $H$, a spanning sub-hypergraph $H'$ of $H$ is $\mathcal{F}$-saturated in $H$ if $H'$ is $\mathcal{F}$-free, but adding any edge in $E(H)\backslash E(H')$ to $H'$ creates a copy of some $F\in\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges in an $\mathcal{F}$-saturated spanning sub-hypergraph of $H$. In this paper, we asymptotically determine the saturation number of Berge stars in random $r$-uniform hypergraphs.

scholarly journals Minimal TUD spaces

2002 ◽  
Vol 3 (1) ◽  
pp. 55 ◽  
Author(s):  
A.E. McCluskey ◽  
W.S. Watson
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Linear Order ◽  
Closed Sets ◽  
Separation Axioms ◽  
Derived Set ◽  
Weak Separation

<p>A topological space is T<sub>UD</sub> if the derived set of each point is the union of disjoint closed sets. We show that there is a minimal T<sub>UD</sub> space which is not just the Alexandroff topology on a linear order. Indeed the structure of the underlying partial order of a minimal T<sub>UD</sub> space can be quite complex. This contrasts sharply with the known results on minimality for weak separation axioms.</p>

Erdős–Ko–Rado in Random Hypergraphs

2009 ◽  
Vol 18 (5) ◽  
pp. 629-646 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
TOM BOHMAN ◽  
DHRUV MUBAYI
Keyword(s):  
Maximum Size ◽  
Projective Planes ◽  
Uniform Hypergraph ◽  
Open Questions ◽  
Wide Range ◽  
Random Hypergraphs ◽  
The Way

Let 3 ≤k<n/2. We prove the analogue of the Erdős–Ko–Rado theorem for the randomk-uniform hypergraphGk(n,p) whenk< (n/2)1/3; that is, we show that with probability tending to 1 asn→ ∞, the maximum size of an intersecting subfamily ofGk(n,p) is the size of a maximum trivial family. The analogue of the Erdős–Ko–Rado theorem does not hold for allpwhenk≫n1/3.We give quite precise results fork<n1/2−ϵ. For largerkwe show that the random Erdős–Ko–Rado theorem holds as long aspis not too small, and fails to hold for a wide range of smaller values ofp. Along the way, we prove that every non-trivial intersectingk-uniform hypergraph can be covered byk2−k+ 1 pairs, which is sharp as evidenced by projective planes. This improves upon a result of Sanders [7]. Several open questions remain.

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