scholarly journals On Kruskal’s theorem that every 3 × 3 × 3 array has rank at most 5

2013 ◽  
Vol 439 (2) ◽  
pp. 401-421 ◽  
Author(s):  
Murray R. Bremner ◽  
Jiaxiong Hu
Keyword(s):  
Kruskal’S Theorem

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.

Extensions of arithmetic for proving termination of computations

1989 ◽  
Vol 54 (3) ◽  
pp. 779-794 ◽  
Author(s):  
Clement F. Kent ◽  
Bernard R. Hodgson
Keyword(s):  
Wide Class ◽  
Combinatorial Algorithms ◽  
Kruskal’S Theorem

AbstractKirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are equivalent, over elementary arithmetic, and contrast these results with those recently obtained for Kruskal's theorem.

scholarly journals From Kruskal’s theorem to Friedman’s gap condition

2020 ◽  
Vol 30 (8) ◽  
pp. 952-975
Author(s):  
Anton Freund
Keyword(s):  
Mathematical Logic ◽  
Building Blocks ◽  
Graph Minor ◽  
Kruskal’S Theorem ◽  
Independence Results ◽  
Labelled Trees

AbstractHarvey Friedman’s gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: It arises via iterated applications of a uniform Kruskal theorem.

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