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The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (); a short proof was given by Crispin Nash-Williams ().It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR 0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected , it finds a minimum spanning tree . It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle . [ 2 ]
Krull–Schmidt theorem (group theory) Kruskal's tree theorem (order theory) Kruskal–Katona theorem (combinatorics) Krylov–Bogolyubov theorem (dynamical systems) Kuhn's theorem (game theory) Kuiper's theorem (operator theory, topology) Künneth theorem (algebraic topology) Kurosh subgroup theorem (group theory) Kutta–Joukowski theorem
Pages in category "Theorems in discrete mathematics" The following 42 pages are in this category, out of 42 total. ... Kruskal's tree theorem; L. Large set (Ramsey ...
Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
In combinatorics, he is known for Kruskal's tree theorem (1960), which is also interesting from a mathematical logic perspective since it can only be proved nonconstructively. Kruskal also applied his work in linguistics , in an experimental lexicostatistical study of Indo-European languages , together with the linguists Isidore Dyen and Paul ...
For example, [Friedman, Harvey (1988), "The Incompleteness Phenomena", Proceedings of the AMS Centennial Celebration, August 8-12, 1988, AMS Centennial Publications, Volume II, Mathematics into the Twenty-first Century, 1992, pp. 73-79] discusses Kruskal's Tree Theorem under three different definitions of the term "tree": trees as partially ...
For example, the statement "there is an integer n such that if there is a sequence of rooted trees T 1, T 2, ..., T n such that T k has at most k+10 vertices, then some tree can be homeomorphically embedded in a later one" is provable in Peano arithmetic, but the shortest proof has length at least 1000 2, where 0 2 = 1 and n + 1 2 = 2 (n 2 ...