By F. P. Ramsey (auth.), Ira Gessel, Gian-Carlo Rota (eds.)

This quantity surveys the advance of combinatorics due to the fact 1930 by means of featuring in chronological order the basic result of the topic proved in over 5 many years of unique 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.-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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NI + N, + ... + N,... R, ~ N = Let G ' be G separated into its components, and let R' be the rank of G' . G is formed from G' by letting vertices of different components coalesce. Each time we join two pieces, the number of vertices and the number of connected pieces arc each reduced by 1, so that the rank remains the same. Thus R = R' . Now 1" = 1'I + V t +···+ V ... , 1" = P I + P2 + ... + p ... (where each P i = 1). Subtracting, R = R' = RI + R2 + ... + R ... E + E, + ... + E. , As also ~ E, it follows that N = NI + Nt + ...

M. Then G' is a dual ofG. Let H be any subgraph of G, and let the parts of H in GI, ... , Gm be HI, ... , Hm. Then H' is the complement of the subgraph in G' corresponding to H in G. Using the proof of Theorem 13, we find that 1" + ... + 1',,{ , = 1'{ and As also R' = R{ and 1'f + ... + R,,{ = Rf - ni (i = 1, ... , m), adding these last equations gives 1" = R' - n, and hence G' is a dual of G. THEOREM 24. Let GI, ... , Gm and G{, ... , G"{ be the components of the dual graphs G and G', and let the correspondence between these two graphs be such that arcs in G; correspond to arcs in GI, i = 1, ...

Individuals. present our condition in a more striking • We have shown this when r < ,,; we may also have r = n, but then there is nothing to prove since in a function of It variables every alternative is serial. EM OF FORMAL LOGIC. It is necessary and sufficient for the consistency of the formula. y = y. (1) The universal function x (2) The null function x =1= x . y =1= y. = y. (3) Identity x (4) Difference x =1= y. e. satisfying (a) (x) -- ¢(x, x), (b) (x, y)[x = y V {p(x, y). "" p(y, x) f V {rp(y, x).