Classic Papers in Combinatorics

Classic Papers in Combinatorics

Language: English

Pages: 492

ISBN: 0817633642

Format: PDF / Kindle (mobi) / ePub


This volume surveys the development of combinatorics since 1930 by presenting in chronological order the fundamental results of the subject proved in over five decades of original papers by: T. van Aardenne-Ehrenfest.- R.L. Brooks.- N.G. de Bruijn.- G.F. Clements.- H.H. Crapo.- R.P. Dilworth.- J. Edmonds.- P. Erdös.- L.R. Ford, Jr.- D.R. Fulkerson.- D. Gale.- L. Geissinger.- I.J. Good.- R.L. Graham.- A.W. Hales.- P. Hall.- P.R. Halmos.- R.I. Jewett.- I. Kaplansky.- P.W. Kasteleyn.- G. Katona.- D.J. Kleitman.- K. Leeb.- B. Lindström.- L. Lovász.- D. Lubell.- C. St. J.A. Nash-Williams.- G. Pólya.-R. Rado.- F.P. Ramsey.- G.-C. Rota.- B.L. Rothschild.- H.J. Ryser.- C. Schensted.- M.P. Schützenberger.- R.P. Stanley.- G. Szekeres.- W.T. Tutte.- H.E. Vaughan.- H. Whitney.

Algorithms

Understanding Complex Datasets: Data Mining with Matrix Decompositions (Chapman & Hall/CRC Data Mining and Knowledge Discovery Series)

Introduction to Linear Algebra - Instructor's Manual (3rd Edition)

Descartes on Polyhedra: A Study of the De Solidorum Elementis (Sources in the History of Mathematics and Physical Sciences)

Mathematical Analysis I (UNITEXT, Volume 84) (2nd Edition)

A Course in Formal Languages, Automata and Groups (Universitext)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

characterizing perfect graphs. Philip Hall's paper [1935b] was the first in what is now called matching theory. A very short proof of Hall's marriage theorem was given by Halmos and Vaughan [1950b]. In the same year Dilworth [l950a] proved his famous decomposition theorem for partially ordered sets, which generalizes Hall's theorem. Several other minimax combinatorial theorems can be viewed as variants or generalizations of the marriage theorem. Such are Tutte's definitive work on factors in

k(a). Then D is a sublattice of a direct union of k chains and k is the smallest number for which such an imbedding holds. 2. Proof of Theorem 1.1. We shall prove the theorem first for the case where P is finite. The theorem in the general case will then follow by a transfinite argument. Hence let P be a finite partially ordered set and let k be the maximal number of independent elements. If k = 1, then every two elements of P are comparable and P is thus 1 This theorem has a certain formal

and if some set of h baclwlor, were to know fewer than h spinsters, then this set of h bachelors togdher with the k married men would have known fewer than k h girls. .\n + + " R('('eiH'd .Jllne (j. l!H!l. H. ".('~.l. "_-\llI1o~t pl'rioclie inn1l'iant "edor ~ets in a metric "ector Sp:l'·~." ..tmcrirfl" .'ounlal of J{flth('mfltic.~. yol. i1 (l!l4!)), PI'. li8-20;;. 2 P. Hall. "On ]'('prpsl'ntation of sllh~et<' ./ounlal of the LOlldon Jfathcm(JI;,nl Society. yol. 10 (l!):l;'). pp. 2(j·:l0. 3 C

l' and its contradictory ....., l' for sets of arguments which become the same when Xi is X2' (x 1 , X 2, xg) . ....., 1'(X2, Xl' xs)]· substituted for Xj [e.g. Xl Any alternative which violates these laws must always be false and can evidently be discarded without affecting the consistency of the formula. The remaining alternatives can then be classified according to the number of x's they make to be different, which lllay be anything from 1 up to n. Suppose that for a given alternative this

would then .. We take one function of one variable only for simplicit,y; also to save space we :>mit expressions which may be taken for granted, such as Xl = X" Xl =1= x,. 12 1928.] Ox 275 A PROBLEM OF FORMAL LOGIC. be true for a certain set of :;r's and F would be false for these :;r's contrary to hypothesis. Hence p is never the true alternative and may be omitted without affecting the consistency of the formula. \Vhen we h:1\"e discarded all these alternatives from F, the remainder will

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