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The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
The Kruskal–Katona theorem relating the size of a family of equal-sized sets and the size of the family of subsets of its sets of a smaller equal size. [2] Cap sets and the sunflower conjecture on families of sets with equal pairwise intersection. [2] Open problems including Frankl's union-closed sets conjecture. [2]
The theorem is named after Joseph Kruskal and Gyula O. H. Katona, who published it in 1963 and 1968 respectively. According to Le & Römer (2019) , it was discovered independently by Kruskal (1963) , Katona (1968) , Marcel-Paul Schützenberger ( 1959 ), Harper (1966) , and Clements & Lindström (1969) .
Kruskal's tree theorem states that, in every infinite set of finite trees, there exists a pair of trees one of which is homeomorphically embedded into the other; another way of stating the same fact is that the homeomorphisms of trees form a well-quasi-ordering.
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 ]
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
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.
Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system ATR 0 codifying the principles acceptable based on a philosophy of mathematics called predicativism. [8]