Colourings of Uniform Hypergraphs with Large Girth and Applications

2017 ◽  
Vol 27 (2) ◽  
pp. 245-273 ◽  
Author(s):  
ANDREY KUPAVSKII ◽  
DMITRY SHABANOV
Keyword(s):  
Lower Bound ◽  
Arithmetic Progression ◽  
Absolute Constant ◽  
Combinatorial Problem ◽  
Uniform Hypergraphs ◽  
Probabilistic Technique ◽  
Formula Group ◽  
Large Girth ◽  
Image Position

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that ifHis ann-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality$$ \Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n} $$for some absolute constantc> 0.As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden numberW(n, r), the minimum naturalNsuch that in an arbitrary colouring of the set of integers {1,. . .,N} withrcolours there exists a monochromatic arithmetic progression of lengthn. We prove that$$ W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}. $$

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.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

scholarly journals Normal Numbers and the Normality Measure

2013 ◽  
Vol 22 (3) ◽  
pp. 342-345 ◽  
Author(s):  
CHRISTOPH AISTLEITNER
Keyword(s):  
Lower Bound ◽  
Binary Sequence ◽  
Normal Numbers ◽  
Small Discrepancy ◽  
Normal Number ◽  
Formula Group ◽  
Image Position ◽  
Normality Measure

In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira and Rödl proved that the minimal possible value of the normality measure of an N-element binary sequence satisfies \begin{equation*} \biggl( \frac{1}{2} + o(1) \biggr) \log_2 N \leq \min_{E_N \in \{0,1\}^N} \mathcal{N}(E_N) \leq 3 N^{1/3} (\log N)^{2/3} \end{equation*} for sufficiently large N, and conjectured that the lower bound can be improved to some power of N. In this note it is observed that a construction of Levin of a normal number having small discrepancy gives a construction of a binary sequence EN with (EN) = O((log N)2), thus disproving the conjecture above.

scholarly journals Arithmetic Progressions in Sumsets and Lp-Almost-Periodicity

2013 ◽  
Vol 22 (3) ◽  
pp. 351-365 ◽  
Author(s):  
ERNIE CROOT ◽  
IZABELLA ŁABA ◽  
OLOF SISASK
Keyword(s):  
Arithmetic Progression ◽  
Short Proof ◽  
Arithmetic Progressions ◽  
Almost Periodicity ◽  
General Lemma ◽  
Formula Group ◽  
Alpha Beta ◽  
Image Position ◽  
Sum Of Functions

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least \begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation} Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least \begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation}

scholarly journals Combinatorial Structures on van der Waerden sets

2015 ◽  
Vol 24 (6) ◽  
pp. 929-953
Author(s):  
KONSTANTINOS TYROS
Keyword(s):  
Level Set ◽  
Arithmetic Progression ◽  
Positive Density ◽  
Homogeneous Tree ◽  
Positive Real ◽  
Combinatorial Structures ◽  
Abstract Setting ◽  
Formula Group ◽  
Image Position ◽  
Positive Integers

In this paper we provide two results. The first one consists of an infinitary version of the Furstenberg–Weiss theorem. More precisely we show that every subsetAof a homogeneous treeTsuch that$\frac{|A\cap T(n)|}{|T(n)|}\geqslant\delta,$whereT(n) denotes thenth level ofT, for allnin a van der Waerden set, for some positive real δ, contains a strong subtree having a level set which forms a van der Waerden set.The second result is the following. For every sequence (mq)q∈ℕof positive integers and for every real 0 < δ ⩽ 1, there exists a sequence (nq)q∈ℕof positive integers such that for everyD⊆ ∪k∏q=0k-1[nq] satisfying$\frac{\big|D\cap \prod_{q=0}^{k-1} [n_q]\big|s}{\prod_{q=0}^{k-1}n_q}\geqslant\delta$for everykin a van der Waerden set, there is a sequence (Jq)q∈ℕ, whereJqis an arithmetic progression of lengthmqcontained in [nq] for allq, such that ∏q=0k-1Jq⊆Dfor everykin a van der Waerden set. Moreover, working in an abstract setting, we may requireJqto be any configuration of natural numbers that can be found in an arbitrary set of positive density.

scholarly journals Multi-crossing number for knots and the Kauffman bracket polynomial

2016 ◽  
Vol 164 (1) ◽  
pp. 147-178 ◽  
Author(s):  
COLIN ADAMS ◽  
ORSOLA CAPOVILLA-SEARLE ◽  
JESSE FREEMAN ◽  
DANIEL IRVINE ◽  
SAMANTHA PETTI ◽  
...  
Keyword(s):  
Singular Point ◽  
Lower Bound ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Kauffman Bracket ◽  
Classic Result ◽  
Formula Group ◽  
Image Position ◽  
Bracket Polynomial ◽  
Extensive List

AbstractA multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.

scholarly journals Largest Components in Random Hypergraphs

2018 ◽  
Vol 27 (5) ◽  
pp. 741-762 ◽  
Author(s):  
OLIVER COOLEY ◽  
MIHYUN KANG ◽  
YURY PERSON
Keyword(s):  
Connected Components ◽  
Bounded Degree ◽  
Uniform Hypergraphs ◽  
Original Contribution ◽  
Common Notion ◽  
Formula Group ◽  
The Common ◽  
Edge Probability ◽  
Image Position ◽  
Random Hypergraphs

In this paper we considerj-tuple-connected components in randomk-uniform hypergraphs (thej-tuple-connectedness relation can be defined by letting twoj-sets be connected if they lie in a common edge and considering the transitive closure; the casej= 1 corresponds to the common notion of vertex-connectedness). We show that the existence of aj-tuple-connected component containing Θ(nj)j-sets undergoes a phase transition and show that the threshold occurs at edge probability$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.Our main original contribution is abounded degree lemma, which controls the structure of the component grown in the search process.

scholarly journals Saturated Subgraphs of the Hypercube

2016 ◽  
Vol 26 (1) ◽  
pp. 52-67 ◽  
Author(s):  
J. ROBERT JOHNSON ◽  
TREVOR PINTO
Keyword(s):  
Lower Bound ◽  
Constant Factor ◽  
Formula Group ◽  
Minimum Number ◽  
Image Position

We say a graph is (Qn,Qm)-saturatedif it is a maximalQm-free subgraph of then-dimensional hypercubeQn. A graph is said to be (Qn,Qm)-semi-saturatedif it is a subgraph ofQnand adding any edge forms a new copy ofQm. The minimum number of edges a (Qn,Qm)-saturated graph (respectively (Qn,Qm)-semi-saturated graph) can have is denoted by sat(Qn,Qm) (respectivelys-sat(Qn,Qm)). We prove that$$ \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath}$$for fixedm, disproving a conjecture of Santolupo that, whenm=2, this limit is 1/4. Further, we show by a different method that sat(Qn,Q2)=O(2n), and thats-sat(Qn,Qm)=O(2n), for fixedm. We also prove the lower bound$$ \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath}$$thus determining sat(Qn,Q2) to within a constant factor, and discuss some further questions.

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.

NORM OF THE HILBERT MATRIX OPERATOR ON THE WEIGHTED BERGMAN SPACES

2017 ◽  
Vol 60 (3) ◽  
pp. 513-525 ◽  
Author(s):  
BOBAN KARAPETROVIĆ
Keyword(s):  
Lower Bound ◽  
Bergman Space ◽  
Upper Bound ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Matrix Operator ◽  
Weighted Bergman Spaces ◽  
Formula Group ◽  
Hilbert Matrix ◽  
Image Position

AbstractWe find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap,α \begin{equation*} \|H\|_{A^{p,\alpha}\rightarrow A^{p,\alpha}}\geq\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}, \,\, \textnormal{for} \,\, 1<\alpha+2<p. \end{equation*} We show that if 4 ≤ 2(α + 2) ≤ p, then ∥H∥Ap,α → Ap,α = $\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$, while if 2 ≤ α +2 < p < 2(α+2), upper bound for the norm ∥H∥Ap,α → Ap,α, better then known, is obtained.

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