On the Independence Number of Steiner Systems

2013 ◽  
Vol 22 (2) ◽  
pp. 241-252 ◽  
Author(s):  
ALEX EUSTIS ◽  
JACQUES VERSTRAËTE
Keyword(s):  
Independence Number ◽  
Maximum Size ◽  
Uniform Hypergraph ◽  
Steiner Systems ◽  
L System ◽  
Formula Group ◽  
Image Position

Apartial Steiner (n,r,l)-systemis anr-uniform hypergraph onnvertices in which every set oflvertices is contained in at most one edge. A partial Steiner (n,r,l)-system iscompleteif every set oflvertices is contained in exactly one edge. In a hypergraph, the independence number α() denotes the maximum size of a set of vertices incontaining no edge. In this article we prove the following. Given integersr,lsuch thatr≥ 2l− 1 ≥ 3, we prove that there exists a partial Steiner (n,r,l)-systemsuch that$$\alpha(\HH) \lesssim \biggl(\frac{l-1}{r-1}(r)_l\biggr)^{\frac{1}{r-1}}n^{\frac{r-l}{r-1}} (\log n)^{\frac{1}{r-1}} \quad \mbox{ as }n \rightarrow \infty.$$This improves earlier results of Phelps and Rödl, and Rödl and Ŝinajová. We conjecture that it is best possible as it matches the independence number of a randomr-uniform hypergraph of the same density. Ifl= 2 orl= 3, then for infinitely manyrthe partial Steiner systems constructed are complete for infinitely manyn.

scholarly journals On the Normalized Shannon Capacity of a Union

2016 ◽  
Vol 25 (5) ◽  
pp. 766-767 ◽  
Author(s):  
PETER KEEVASH ◽  
EOIN LONG
Keyword(s):  
Independence Number ◽  
Shannon Capacity ◽  
Strong Product ◽  
Formula Group ◽  
Image Position ◽  
Product Of Graphs

Let G1 × G2 denote the strong product of graphs G1 and G2, that is, the graph on V(G1) × V(G2) in which (u1, u2) and (v1, v2) are adjacent if for each i = 1, 2 we have ui = vi or uivi ∈ E(Gi). The Shannon capacity of G is c(G) = limn → ∞ α(Gn)1/n, where Gn denotes the n-fold strong power of G, and α(H) denotes the independence number of a graph H. The normalized Shannon capacity of G is $$C(G) = \ffrac {\log c(G)}{\log |V(G)|}.$$ Alon [1] asked whether for every ε < 0 there are graphs G and G′ satisfying C(G), C(G′) < ε but with C(G + G′) > 1 − ε. We show that the answer is no.

Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

2017 ◽  
Vol 27 (3) ◽  
pp. 358-386 ◽  
Author(s):  
CLAYTON COLLIER-CARTAINO ◽  
NATHAN GRABER ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Positive Constant ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Formula Group ◽  
Image Position ◽  
Linear Hypergraphs

Anr-uniform hypergraph is called anr-graph. A hypergraph islinearif every two edges intersect in at most one vertex. Given a linearr-graphHand a positive integern, thelinear Turán numberexL(n,H) is the maximum number of edges in a linearr-graphGthat does not containHas a subgraph. For each ℓ ≥ 3, letCrℓdenote ther-uniform linear cycle of length ℓ, which is anr-graph with edgese1, . . .,eℓsuch that, for alli∈ [ℓ−1], |ei∩ei+1|=1, |eℓ∩e1|=1 andei∩ej= ∅ for all other pairs {i,j},i≠j. For allr≥ 3 and ℓ ≥ 3, we show that there exists a positive constantc=cr,ℓ, depending onlyrand ℓ, such that exL(n,Crℓ) ≤cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constantsa=am,randb=bm,r, depending only onmandr, such that\begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

Turán-Ramsey Theorems and Kp-Independence Numbers

1994 ◽  
Vol 3 (3) ◽  
pp. 297-325 ◽  
Author(s):  
P. Erdős ◽  
A. Hajnal ◽  
M. Simonovits ◽  
V. T. Sós ◽  
E. Szemerédi
Keyword(s):  
Simple Structure ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Asymptotic Value ◽  
Image Position ◽  
Independence Numbers ◽  
Edge Colouring

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value offor p < q. Several results are proved, and some other problems and conjectures are stated.

scholarly journals Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

2017 ◽  
Vol 26 (3) ◽  
pp. 406-422
Author(s):  
MARCELO M. GAUY ◽  
HIÊP HÀN ◽  
IGOR C. OLIVEIRA
Keyword(s):  
Lower Bound ◽  
Uniform Hypergraph ◽  
Small Interval ◽  
Asymptotic Size ◽  
Intersecting Family ◽  
Formula Group ◽  
Image Position ◽  
Random Hypergraphs

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$k(n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size $$(1+o(1))p\ffrac kn N$$ for any $$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well.A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D−1 ≪ p ⩽ (n/k)1−ϵD−1, the largest intersecting subhypergraph of $\mathcal{H}$k(n, p) has size Θ(ln(pD)ND−1), provided that $k \gg \sqrt{n \ln n}$.Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$k, for essentially all values of p and k.

Multicolour Sunflowers

2018 ◽  
Vol 27 (6) ◽  
pp. 974-987
Author(s):  
DHRUV MUBAYI ◽  
LUJIA WANG
Keyword(s):  
Maximum Size ◽  
Fixed Size ◽  
Common Intersection ◽  
Formula Group ◽  
The Common ◽  
Image Position ◽  
Families Of Subsets

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersectionCof all of them, and |C| is smaller than each of the sets. A longstanding conjecture due to Erdős and Szemerédi (solved recently in [7, 9]; see also [22]) was that the maximum size of a family of subsets of [n] that contains no sunflower of fixed sizek> 2 is exponentially smaller than 2nasn→ ∞. We consider the problems of determining the maximum sum and product ofkfamilies of subsets of [n] that contain no sunflower of sizekwith one set from each family. For the sum, we prove that the maximum is$$(k-1)2^n+1+\sum_{s=0}^{k-2}\binom{n}{s}$$for alln⩾k⩾ 3, and for thek= 3 case of the product, we prove that the maximum is$$\biggl(\ffrac{1}{8}+o(1)\biggr)2^{3n}.$$We conjecture that for all fixedk⩾ 3, the maximum product is (1/8+o(1))2kn.

Cliques in Steiner systems

2009 ◽  
Vol 59 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
František Franěk ◽  
Vojtěch Rödl
Keyword(s):  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Steiner Systems ◽  
Lower And Upper Bounds ◽  
Element Subset ◽  
L System

AbstractA partial Steiner (k,l)-system is a k-uniform hypergraph with the property that every l-element subset of V is contained in at most one edge of . In this paper we show that for given k,l and t there exists a partial Steiner (k,l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of this note, we establish asymptotic lower and upper bounds on the size of such cliques with respect to the order of Steiner systems.

scholarly journals Packing Loose Hamilton Cycles

2017 ◽  
Vol 26 (6) ◽  
pp. 839-849
Author(s):  
ASAF FERBER ◽  
KYLE LUH ◽  
DANIEL MONTEALEGRE ◽  
OANH NGUYEN
Keyword(s):  
Packing Problem ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Threshold Probability ◽  
Edge Disjoint ◽  
Formula Group ◽  
Vertex Set ◽  
Substantial Progress ◽  
Image Position

A subsetCof edges in ak-uniform hypergraphHis aloose Hamilton cycleifCcovers all the vertices ofHand there exists a cyclic ordering of these vertices such that the edges inCare segments of that order and such that every two consecutive edges share exactly one vertex. The binomial randomk-uniform hypergraphHkn,phas vertex set [n] and an edge setEobtained by adding eachk-tuplee∈ ($\binom{[n]}{k}$) toEwith probabilityp, independently at random.Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all buto(|E|) edges, referred to as thepacking problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle inHkn,pis$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$the best known bounds for the packing problem are aroundp= polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: forp≥ logCn/nk−1, a randomk-uniform hypergraphHkn,pwith high probability contains$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$edge-disjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

2016 ◽  
Vol 59 (3) ◽  
pp. 533-547 ◽  
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
General Setting ◽  
Maximal Operators ◽  
List Type ◽  
Type Number ◽  
Type Estimate ◽  
Formula Group ◽  
Probability Spaces ◽  
Image Position

AbstractLet $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

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