Classic Papers in Combinatorics (Modern Birkhäuser Classics) by Gian-Carlo Rota, Ira Gessel

By Gian-Carlo Rota, Ira Gessel

This quantity surveys the improvement of combinatorics seeing that 1930 via featuring in chronological order the basic result of the topic proved in over 5 a long time of unique papers through: 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.- okay. 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.

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Thus the increase in the number of connected pieces in G' is just N (hence, in particular, G' itself is connected) and therefore this number increases only when the nullity of G increases, which proves (2). Let now H be any subgraph of G, let H' be the complement of the corresponding subgraph of G', and let H' include all the vertices of G'. We build up H arc by arc, at the same time removing the corresponding arcs of G'. Thus when H is formed, H' also is formed. By (1) and (2), the increase in the number of connected pieces in forming H' from G' equals the nullity of H, that is, But p' - p' r' = V' - = n.

Suppose that by this process G turns into L(CP, x' y" ... , CPo, XO' "', p, q, ... , =, Xl' ,1:2' ••• , X,,). Then the consistency of formula (i) in a universe of N individuals is evidently equivalent to the consistency in a universe of N - p. individuals of the formula (Xl' X 2, ••• , x,,)L(CP, X, y" ... , CPo, XO' ••• , p, q, ... , =, Xl' x~, ••• , Xn). But this is a formula of the type previously dealt with, except for the variable propositions p, q, ... , which are easily eliminated by considering the different cases of their truth and falsity, the formula being consistent jf it is cOl1sist-ent in one snch casco Reprinted from Proc.

We take P will be false unless it completely contains a serial A". For thip, it is enongh to show that, given any 1'0, Xo, ~o, ... , there must be a set of ny's for which the true form is serial *, or, since a serial form is one which contains a serial alternative, that there must be a set of values of 'VI, Y2, ... , Yn for which the true alternative is serial. Let us suppose that among our functions 1'0, Xo, 'fro, ... there are aJ functions of one variable, U2 of two variables, ... , and G" of n variables, and let us order the universe by a serial relation R.

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