On well-quasi-ordering infinite trees – Nash–Williams's theorem revisited

2001 ◽  
Vol 130 (3) ◽  
pp. 401-408 ◽  
Author(s):  
DANIELA KÜHN
Keyword(s):  
Short Proof ◽  
Infinite Trees ◽  
Quasi Ordering

Nash–Williams proved that the infinite trees are well-quasi-ordered (indeed, better-quasi-ordered) under the topological minor relation. We combine ideas of several authors into a more accessible and essentially self-contained short proof.

On well-quasi-ordering infinite trees

1965 ◽  
Vol 61 (3) ◽  
pp. 697-720 ◽  
Author(s):  
C. St. J. A. Nash-Williams
Keyword(s):  
Ordered Set ◽  
Infinite Sequence ◽  
Transitive Relation ◽  
Infinite Trees ◽  
Finite Sequences ◽  
Positive Integers ◽  
Quasi Ordering

AbstractLet A be the set of all ascending finite sequences (with at least one term) of positive integers. Let s, t ∈ A. Write s ⊲ t if there exist m, n, x1, …, xn such that m < n and x1 < … < xn and s is x1, …, xm and t is x2, x3, …, xn. Call a subset S of A a P-block if, for every infinite ascending sequence x1, x2, … of positive integers, there exists an m such that x1, …, xm belongs to S. A quasi-ordered set Q (i.e. a set on which a reflexive and transitive relation ≤ is defined) is better-quasi-ordered if, for every P-block S and every function f:S → Q, there exist s, t ∈ S such that s ⊲ t and f(s) ≤ f(t). It is proved that any set of (finite or infinite) trees is better-quasi-ordered if T1 ≤ T2 means that the tree T1 is homeomorphic to a subtree of the tree T2. This establishes a conjecture of J. B.Kruskal that, if T1, T2, … is an infinite sequence of trees, then there exist i, j such that i < j and Ti ≤ Tj.

A note on generalized quasi-contraction

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3091-3093
Author(s):  
Dejan Ilic ◽  
Darko Kocev
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Short Proof

In this paper we give a short proof of the main results of Kumam, Dung and Sitthithakerngkiet (P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A Generalization of Ciric Fixed Point Theorems, FILOMAT 29:7 (2015), 1549-1556).

Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

2021 ◽  
Vol 344 (7) ◽  
pp. 112430
Author(s):  
Johann Bellmann ◽  
Bjarne Schülke
Keyword(s):  
Short Proof ◽  
Kneser Graphs

scholarly journals Strongly perfect claw‐free graphs—A short proof

2021 ◽  
Author(s):  
Maria Chudnovsky ◽  
Cemil Dibek
Keyword(s):  
Short Proof

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Well-quasi-ordering hereditarily finite sets

2013 ◽  
Vol 90 (6) ◽  
pp. 1278-1291 ◽  
Author(s):  
Alberto Policriti ◽  
Alexandru I. Tomescu
Keyword(s):  
Finite Sets ◽  
Quasi Ordering

scholarly journals A short proof of Gamas’s theorem

2009 ◽  
Vol 430 (2-3) ◽  
pp. 791-794 ◽  
Author(s):  
Andrew Berget
Keyword(s):  
Short Proof
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