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

This quantity surveys the improvement of combinatorics considering 1930 through offering in chronological order the basic result of the topic proved in over 5 a long time 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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XlI)]-, J-l •... , IL in which ~ means" if, then" and G(p, ... , xn) = II F1(p, x' ... , =, Zl' ... , Zl" $1' $2' ... g. Xi =1= Xi) not involving any z. N ext we modify G by introducing new functions. f}. p, with arguments some of which are z's and some x's; from 23 ON 286 A PROBLEM OF FORllAL LOGIC. • Values of

Let CI be a chain in Gl joining am and ai, let C2 be a chain in G2 joining al and a2, ... , let Cm be a chain in Gm joining am-I and am. These chains taken together form a cir• This theorem may also be proved easily fratn Theorem 17. 32 1932) NON-SEPARABLE AND PLANAR GRAPHS 347 cuit P passing through all the graphs. Now separate G into its components. By Theorem 11 (see the converse of Theorem 10), P is contained in one of these components. The same is true of each of the graphs GI , G2 , • • • , Gm , and hence these graphs are all contained in the same component.

R' = = R' - n. R' = But as G' is a dual of G, r' These equations give N. The other equation follows when we note that E' =E. THEOREM 21. If G' is a dual of G, then G is a dual of G'. Let H' be any sub graph of G' , and let H be the complement of the corresponding sub graph of G. Then, as G' is a dual of G, * While this definition agrees with the ordinary one for graphs lying on a plane or sphere, a graph on a surface of higher connectivity, such as the torus, has in general no dual. ) 37 352 HASSLER WHITNEY r' = R' - n.