scholarly journals On Erdős–Ko–Rado for Random Hypergraphs II

2018 ◽  
Vol 28 (1) ◽  
pp. 61-80 ◽  
Author(s):  
A. HAMM ◽  
J. KAHN
Keyword(s):  
Sperner's Theorem ◽  
Random Hypergraphs

Denote by ${\mathcal H}_k$(n, p) the random k-graph in which each k-subset of {1,. . .,n} is present with probability p, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed ε > 0 such that if n = 2k + 1 and p > 1 - ε, then w.h.p. (that is, with probability tending to 1 as k → ∞), ${\mathcal H}_k$(n, p) has the ‘Erdős–Ko–Rado property’. We also mention a similar random version of Sperner's theorem.

Panchromatic colorings of random hypergraphs

2021 ◽  
Vol 31 (1) ◽  
pp. 19-41
Author(s):  
D. A. Kravtsov ◽  
N. E. Krokhmal ◽  
D. A. Shabanov
Keyword(s):  
Binomial Model ◽  
Threshold Probability ◽  
The Difference ◽  
Random Hypergraphs

Abstract We study the threshold probability for the existence of a panchromatic coloring with r colors for a random k-homogeneous hypergraph in the binomial model H(n, k, p), that is, a coloring such that each edge of the hypergraph contains the vertices of all r colors. It is shown that this threshold probability for fixed r < k and increasing n corresponds to the sparse case, i. e. the case p = c n / ( n k ) $p = cn/{n \choose k}$ for positive fixed c. Estimates of the form c 1(r, k) < c < c 2(r, k) for the parameter c are found such that the difference c 2(r, k) − c 1(r, k) converges exponentially fast to zero if r is fixed and k tends to infinity.

On Spanning Structures in Random Hypergraphs

2015 ◽  
Vol 49 ◽  
pp. 611-619 ◽  
Author(s):  
O. Parczyk ◽  
Y. Person
Keyword(s):  
Random Hypergraphs

scholarly journals Strong versions of Sperner's theorem

1976 ◽  
Vol 20 (1) ◽  
pp. 80-88 ◽  
Author(s):  
Curtis Greene ◽  
Daniel J Kleitman
Keyword(s):  
Sperner's Theorem

Two-coloring random hypergraphs

2002 ◽  
Vol 20 (2) ◽  
pp. 249-259 ◽  
Author(s):  
Dimitris Achlioptas ◽  
Jeong Han kim ◽  
Michael Krivelevich ◽  
Prasad Tetali
Keyword(s):  
Random Hypergraphs

scholarly journals A generalization of Sperner's theorem

1981 ◽  
Vol 31 (4) ◽  
pp. 481-485 ◽  
Author(s):  
D. E. Daykin ◽  
P. Frankl ◽  
C. Greene ◽  
A. J. W. Hilton
Keyword(s):  
Sperner's Theorem ◽  
Lym Inequality ◽  
Families Of Subsets

AbstractSome generalizations of Sperner's theorem and of the LYM inequality are given to the case when A1,… At are t families of subsets of {1,…,m} such that a set in one family does not properly contain a set in another.

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.

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