Some Remarks on Ramsay's Theorem

1964 ◽  
Vol 7 (4) ◽  
pp. 619-622 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Independent Set ◽  
Infinite Subset ◽  
Infinite Graph ◽  
Complete Subgraph ◽  
Special Case

A special case of a well known theorem of Ramsay [3] states that an infinite graph either contains an infinite complete subgraph or it contains an infinite independent set; in other words there exists an infinite subset of its vertices so that either every two of them are joined by an edge or no two of them are joined by an edge. Thus if we have a graph whose vertices are the integers, and which has no infinite complete sub-graph, it certainly has an infinite independent set. The question can now be asked if there exists an independent set whose vertices n1 < n2 < … do not tend to infinity too fast.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

scholarly journals A Canonical Ramsey Theorem for Exactly m-Coloured Complete Subgraphs

2013 ◽  
Vol 23 (1) ◽  
pp. 102-115 ◽  
Author(s):  
TEERADEJ KITTIPASSORN ◽  
BHARGAV P. NARAYANAN
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Infinite Subset ◽  
Complete Subgraph ◽  
Ramsey Theorem ◽  
Distinguished Vertex ◽  
Edge Colourings ◽  
Distinct Colour

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

scholarly journals Budget Feasible Mechanisms on Matroids

Algorithmica ◽  
2020 ◽  
Author(s):  
Stefano Leonardi ◽  
Gianpiero Monaco ◽  
Piotr Sankowski ◽  
Qiang Zhang
Keyword(s):  
Polynomial Time ◽  
Private Information ◽  
Independent Set ◽  
Deterministic Algorithm ◽  
Matroid Intersection ◽  
Practical Applications ◽  
Incentive Compatible ◽  
The Cost ◽  
Special Case ◽  
Budget Feasible

AbstractMotivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) . Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an $$\alpha $$ α -approximation to the problem of finding a maximum-value independent set in $${\mathcal {M}}$$ M , there exists an individually rational, truthful and budget feasible mechanism which is $$(3\alpha +1)$$ ( 3 α + 1 ) -approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.

scholarly journals On a Conjecture Concerning the Petersen Graph

10.37236/507 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Donald Nelson ◽  
Michael D. Plummer ◽  
Neil Robertson ◽  
Xiaoya Zha
Keyword(s):  
Connected Graph ◽  
Independent Set ◽  
Petersen Graph ◽  
Original Form ◽  
Odd Cycle ◽  
Special Case

Robertson has conjectured that the only 3-connected internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjecture in its original form. In particular, let $G$ be any 3-connected internally-4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If $C$ is any girth cycle in $G$ then $N(C)\backslash V(C)$ cannot be edgeless, and if $N(C) \backslash V(C)$ contains a path of length at least 2, then the conjecture is true. Consequently, if the conjecture is false and $H$ is a counterexample, then for any girth cycle $C$ in $H$, $N(C) \backslash V(C)$ induces a nontrivial matching $M$ together with an independent set of vertices. Moreover, $M$ can be partitioned into (at most) two disjoint non-empty sets where we can precisely describe how these sets are attached to cycle $C$.

scholarly journals Improved bounds for the Erdős-Rogers function

2020 ◽  
Author(s):  
Timothy Gowers ◽  
Oliver Janzer
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Related Problem ◽  
Independent Set ◽  
Large Structure ◽  
Induced Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Subgraph

[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{\alpha_s}$ vertices with $\alpha_s<1/2$ for every $s\ge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2\le t\le 2s-1$ and $s\ge 3$.

Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

2018 ◽  
Vol 41 ◽  
Author(s):  
Daniel Crimston ◽  
Matthew J. Hornsey
Keyword(s):  
General Theory ◽  
Biological Markers ◽  
Natural Environment ◽  
Relevant Dimension ◽  
Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

Test Validation Without Measurement

2016 ◽  
Vol 32 (3) ◽  
pp. 204-214 ◽  
Author(s):  
Emilie Lacot ◽  
Mohammad H. Afzali ◽  
Stéphane Vautier
Keyword(s):  
Test Data ◽  
Test Scores ◽  
Scientific Explanation ◽  
Test Validation ◽  
Clinical Tests ◽  
Ethical Perspective ◽  
Power Of Test ◽  
Memory Accuracy ◽  
Social Uses ◽  
Special Case

Abstract. Test validation based on usual statistical analyses is paradoxical, as, from a falsificationist perspective, they do not test that test data are ordinal measurements, and, from the ethical perspective, they do not justify the use of test scores. This paper (i) proposes some basic definitions, where measurement is a special case of scientific explanation; starting from the examples of memory accuracy and suicidality as scored by two widely used clinical tests/questionnaires. Moreover, it shows (ii) how to elicit the logic of the observable test events underlying the test scores, and (iii) how the measurability of the target theoretical quantities – memory accuracy and suicidality – can and should be tested at the respondent scale as opposed to the scale of aggregates of respondents. (iv) Criterion-related validity is revisited to stress that invoking the explanative power of test data should draw attention on counterexamples instead of statistical summarization. (v) Finally, it is argued that the justification of the use of test scores in specific settings should be part of the test validation task, because, as tests specialists, psychologists are responsible for proposing their tests for social uses.

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