scholarly journals Pattern Avoidance Over a Hypergraph

10.37236/9014 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Benjamin Gunby ◽  
Maxwell Fishelson
Keyword(s):  
Main Tool ◽  
Pattern Avoidance ◽  
Edge Density ◽  
Uniform Hypergraph ◽  
Deterministic Case ◽  
Expected Number ◽  
Classic Result ◽  
Wilf Conjecture ◽  
Random Hypergraph

A classic result of Marcus and Tardos (previously known as the Stanley-Wilf conjecture) bounds from above the number of $n$-permutations ($\sigma \in S_n$) that do not contain a specific sub-permutation. In particular, it states that for any fixed permutation $\pi$, the number of $n$-permutations that avoid $\pi$ is at most exponential in $n$. In this paper, we generalize this result. We bound the number of avoidant $n$-permutations even if they only have to avoid $\pi$ at specific indices. We consider a $k$-uniform hypergraph $\Lambda$ on $n$ vertices and count the $n$-permutations that avoid $\pi$ at the indices corresponding to the edges of $\Lambda$. We analyze both the random and deterministic hypergraph cases. This problem was originally proposed by Asaf Ferber. When $\Lambda$ is a random hypergraph with edge density $\alpha$, we show that the expected number of $\Lambda$-avoiding $n$-permutations is bounded (both upper and lower) as $\exp(O(n))\alpha^{-\frac{n}{k-1}}$, using a supersaturation version of F\"{u}redi-Hajnal. In the deterministic case we show that, for $\Lambda$ containing many size $L$ cliques, the number of $\Lambda$-avoiding $n$-permutations is $O\left(\frac{n\log^{2+\epsilon}n}{L}\right)^n$, giving a nontrivial bound with $L$ polynomial in $n$. Our main tool in the analysis of this deterministic case is the new and revolutionary hypergraph containers method, developed in papers of Balogh-Morris-Samotij and Saxton-Thomason.

scholarly journals Covering and tiling hypergraphs with tight cycles

2020 ◽  
pp. 1-42
Author(s):  
Jie Han ◽  
Allan Lo ◽  
Nicolás Sanhueza-Matamala
Keyword(s):  
Main Tool ◽  
Independent Interest ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Disjoint

Abstract A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$ . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest. For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$ , there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$ , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$ -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

scholarly journals Extremal Problems for $t$-Partite and $t$-Colorable Hypergraphs

10.37236/750 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
John Talbot
Keyword(s):  
Main Tool ◽  
Extremal Problems ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

Fix integers $t \ge r \ge 2$ and an $r$-uniform hypergraph $F$. We prove that the maximum number of edges in a $t$-partite $r$-uniform hypergraph on $n$ vertices that contains no copy of $F$ is $c_{t, F}{n \choose r} +o(n^r)$, where $c_{t, F}$ can be determined by a finite computation. We explicitly define a sequence $F_1, F_2, \ldots$ of $r$-uniform hypergraphs, and prove that the maximum number of edges in a $t$-chromatic $r$-uniform hypergraph on $n$ vertices containing no copy of $F_i$ is $\alpha_{t,r,i}{n \choose r} +o(n^r)$, where $\alpha_{t,r,i}$ can be determined by a finite computation for each $i\ge 1$. In several cases, $\alpha_{t,r,i}$ is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.

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 Unique ergodicity of the horocycle flow on Riemannnian foliations

2018 ◽  
Vol 40 (6) ◽  
pp. 1459-1479
Author(s):  
F. ALCALDE CUESTA ◽  
F. DAL’BO ◽  
M. MARTÍNEZ ◽  
A. VERJOVSKY
Keyword(s):  
Main Tool ◽  
Hyperbolic Surface ◽  
Unique Ergodicity ◽  
Alternative Proof ◽  
Compact Metric Space ◽  
Horocycle Flow ◽  
Full Support ◽  
Classic Result ◽  
Minimal Foliations ◽  
Special Case

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

scholarly journals Absorption probabilities for Gaussian polytopes and regular spherical simplices

2020 ◽  
Vol 52 (2) ◽  
pp. 588-616
Author(s):  
Zakhar Kabluchko ◽  
Dmitry Zaporozhets
Keyword(s):  
Distribution Function ◽  
Main Tool ◽  
Standard Normal Distribution ◽  
Normal Vector ◽  
Normal Distribution Function ◽  
Standard Normal Distribution Function ◽  
Expected Number ◽  
Standard Normal ◽  
Crofton Formula ◽  
Absorption Probabilities

AbstractThe Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

scholarly journals Turán Problems on Non-Uniform Hypergraphs

10.37236/3901 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
J. Travis Johnston ◽  
Linyuan Lu
Keyword(s):  
Recent Work ◽  
Blow Up ◽  
Uniform Hypergraph ◽  
Expected Number ◽  
Uniform Hypergraphs ◽  
Turan Problems ◽  
Vertex Set ◽  
Lubell Function

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This concept, which generalizes  the Turán density of $k$-uniform hypergraphs, is motivated by recent work on extremal poset problems.  The details connecting these two areas will be revealed in the end of this paper.Several properties of Turán density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Turán densities of $\{1,2\}$-hypergraphs.

scholarly journals Triforce and corners

2019 ◽  
Vol 169 (1) ◽  
pp. 209-223
Author(s):  
JACOB FOX ◽  
ASHWIN SAH ◽  
MEHTAAB SAWHNEY ◽  
DAVID STONER ◽  
YUFEI ZHAO
Keyword(s):  
Recent Result ◽  
Higher Dimensions ◽  
The Other ◽  
Edge Density ◽  
Uniform Hypergraph ◽  
Other Hand

AbstractMay the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

scholarly journals The $t$-Stability Number of a Random Graph

10.37236/331 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Ross J. Kang ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Maximum Degree ◽  
Asymptotic Expression ◽  
Main Tool ◽  
Stable Set ◽  
Stable Sets ◽  
Stability Number ◽  
Expected Number ◽  
Maximum Order ◽  
Edge Probability

Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called $t$-stable (or $t$-dependent) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The $t$-stability number $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The theme of this paper is the typical values that this parameter takes on a random graph on $n$ vertices and edge probability equal to $p$. For any fixed $0 < p < 1$ and fixed non-negative integer $t$, we show that, with probability tending to $1$ as $n\to\infty$, the $t$-stability number takes on at most two values which we identify as functions of $t$, $p$ and $n$. The main tool we use is an asymptotic expression for the expected number of $t$-stable sets of order $k$. We derive this expression by performing a precise count of the number of graphs on $k$ vertices that have maximum degree at most $t$.

scholarly journals On the Number of Solutions in Random Hypergraph 2-Colouring

10.37236/6029 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Felicia Rassmann
Keyword(s):  
Direct Consequence ◽  
Limiting Distribution ◽  
Uniform Hypergraph ◽  
Number Of Solutions ◽  
Random Hypergraph ◽  
Density Regime

We determine the limiting distribution of the logarithm of the number of satisfying assignments in the random $k$-uniform hypergraph 2-colouring problem in a certain density regime for all $k\ge 3$. As a direct consequence we obtain that in this regime the random colouring model is contiguous wrt. the planted model, a result that helps simplifying the transfer of statements between these two models.  

scholarly journals Neutrosophic Event-Based Queueing Model

2020 ◽  
pp. 48-55
Author(s):  
Mohamed Bisher Zeina ◽  
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Communication Systems ◽  
Queueing Systems ◽  
Real Life ◽  
Main Tool ◽  
Expected Number ◽  
Mean Waiting Time ◽  
Event Based ◽  
Number Of Customers

In this paper we have defined the concept of neutrosophic queueing systems and defined its neutrosophic performance measures. An important application of neutrosophic logic in queueing systems we face in real life were discussed, that is the neutrosophic events accuring times, because of its wide applications in networking and simulating communication systems specialy when probability distribution is not known, and because it’s more realistic to consider and to not ignore the imprecise events times. Event-based table of a neutrosophic queueing system was presented and its neutrosophic performance measures were derived, i.e. neutrosophic mean waiting time in queue, neutrosophic mean waiting time in system, neutrosophic expected number of customers in queue and neutrosophic expected number of customers in system. Neutrosophic Little’s Formulas (NLF) were also defined which is a main tool in queueing systems problems to make it easier finding performance measures from each other.

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