scholarly journals On the Weight of Berge-$F$-Free Hypergraphs

10.37236/8504 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Sean English ◽  
Dániel Gerbner ◽  
Abhishek Methuku ◽  
Cory Palmer
Keyword(s):  
Short Note ◽  
Ramsey Number ◽  
Weight Functions ◽  
Uniform Hypergraph ◽  
The Way

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$. Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$, the weight of $\mathcal{H}$ is $o(n^2)$. This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Grósz, Methuku and Tompkins.

A Short Note on the Knossos Statuette Inscription

Heritage ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 3186-3192
Author(s):  
Len Gleeson
Keyword(s):  
Short Note ◽  
Evidence Based ◽  
Optical Profilometry ◽  
Physical Evidence ◽  
The Way

For the inscription of the Egyptian statuette in the Heraklion Archaeological Museum, the dedicator’s second title has long been open to question. New and detailed physical evidence, based on optical profilometry, is presented here. The results show errors/omissions in the previously accepted reading and open the way to a much more plausible translation.

Romeinen 9:16, ‘...τρέχοντος...’: Een korte exegetische overweging bij een woord van Paulus

2013 ◽  
Vol 67 (4) ◽  
pp. 303-307
Author(s):  
Tjitze Baarda
Keyword(s):  
Short Note ◽  
Christian Life ◽  
The Law ◽  
Usual Interpretation ◽  
The Way

The usual interpretation of the verb ‘...τρέχοντος...’ assumes that the idea of ‘running’ is derived from the Hellenistic metaphor of foot-races in the stadium, as found elsewhere in the Pauline Letters. It is, of course, quite possible that the original addressees would have understood it this way, but one might ask whether Paul himself was thinking here of runners on the racetrack. One of the issues in the Letter to the Romans is the meaning of the Law in Christian life. It struck me that the poet of the Ode to the Law wrote, ‘I will run in the way of your commandments’ (Ps. 119:32). My intention in this short note is to offer the reader the opportunity to consider whether Paul could have in mind the expression ‘to run’ from this Psalm.

scholarly journals A Note on Odd Cycle-Complete Graph Ramsey Numbers

10.37236/1662 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Benny Sudakov
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Odd Cycle

The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.

scholarly journals Ramsey numbers for certain k-graphs

1981 ◽  
Vol 30 (3) ◽  
pp. 284-296 ◽  
Author(s):  
Stefan A. Burr ◽  
Richard A. Duke
Keyword(s):  
Lower Bounds ◽  
Complete Solution ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Uniform Hypergraph ◽  
Steiner Systems ◽  
Ramsey Numbers ◽  
Existence Question

AbstractWe are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.

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.

scholarly journals On Berge-Ramsey Problems

10.37236/8775 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Dániel Gerbner
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Constant Term ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Large Margin

Given a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ if $V(G)\subset V(\mathcal{H})$ and there is a bijection $f:E(G)\rightarrow E(\mathcal{H})$ such that for any edge $e$ of $G$ we have $e\subset f(e)$. We study Ramsey problems for Berge copies of graphs, i.e. the smallest number of vertices of a complete $r$-uniform hypergraph, such that if we color the hyperedges with $c$ colors, there is a monochromatic Berge copy of $G$. We obtain a couple results regarding these problems. In particular, we determine for which $r$ and $c$ the Ramsey number can be super-linear. We also show a new way to obtain lower bounds, and improve the general lower bounds by a large margin. In the specific case $G=K_n$ and $r=2c-1$, we obtain an upper bound that is sharp besides a constant term, improving earlier results.

scholarly journals On the Multicolor Ramsey Number for 3-Paths of Length Three

10.37236/6670 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Tomasz Łuczak ◽  
Joanna Polcyn
Keyword(s):  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Multicolor Ramsey Number

We show that if we color the hyperedges of the complete $3$-uniform hypergraph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.

On the Erdős–Ginzburg–Ziv invariant and zero-sum Ramsey number for intersecting families

2014 ◽  
Vol 10 (07) ◽  
pp. 1637-1647 ◽  
Author(s):  
Haiyan Zhang ◽  
Guoqing Wang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Zero Sum

Let G be a finite abelian group, and let m > 0 with exp (G) | m. Let sm(G) be the generalized Erdős–Ginzburg–Ziv invariant which denotes the smallest positive integer d such that any sequence of elements in G of length d contains a subsequence of length m with sum zero in G. For any integer r > 0, let [Formula: see text] be the collection of all r-uniform intersecting families of size m. Let [Formula: see text] be the smallest positive integer d such that any G-coloring of the edges of the complete r-uniform hypergraph [Formula: see text] yields a zero-sum copy of some intersecting family in [Formula: see text]. Among other results, we mainly prove that [Formula: see text], where Ω(sm(G)) denotes the least positive integer n such that [Formula: see text], and we show that if r | Ω(sm(G)) – 1 then [Formula: see text].

scholarly journals Note on the Multicolour Size-Ramsey Number for Paths,

10.37236/7954 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Alternative Proof ◽  
Monochromatic Copy

The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = \Omega(r^2 n)$.

scholarly journals On the Generating Hypothesis in Noncommutative Stable Homotopy

2015 ◽  
Vol 116 (2) ◽  
pp. 301 ◽  
Author(s):  
Snigdhayan Mahanta
Keyword(s):  
Recent Work ◽  
Short Note ◽  
Stable Homotopy ◽  
Triangulated Categories ◽  
Generating Hypothesis ◽  
The Way

Freyd's Generating Hypothesis is an important problem in topology with deep structural consequences for finite stable homotopy. Due to its complexity some recent work has examined analogous questions in various other triangulated categories. In this short note we analyze the question in noncommutative stable homotopy, which is a canonical generalization of finite stable homotopy. Along the way we also discuss Spanier-Whitehead duality in this extended setup.

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