Analyse combinatoire, tome 2 by Comtet L.

By Comtet L.

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Chvital, Intersecting families of edges in hypergraphs having the hereditary property, in: C. Berge and D. K. , Hypergraph Seminar 1972, Lecture Notes 41 1 (SpringerVerlag, Berlin, 1972) 61-66. [12] J. Edmonds, Maximum matchings and a polyhedron with (0, 1) vertices, J. Res. Nat. Bur. Standards Ser. B 69 (1965) 125-130; 67-70. [13] P. Erdos, On extremal problems of graphs and generalised graphs, Israel J. Math. 2(1964) 183-190. [14] P. Erdos, Chao-Ko and R. Rado, Intersecting theorems for systems of finite sets, Quart.

Let K - , = 8. (76) For each Jb, let 5: = J - Jb, and (77) Ki = E - cl, (Ja). (78) For each Ki such that K i - l c Ki c cIb(J), let (79) JPtl = {C : C E J, C U e is an Mb-circuit for some e E K,}. We thus generate sequences ? JK ~ -,, c K , c (80) @ = J : c J : c . * * C J " , ~ G J ~ ,J, = J ~ ~ J ~ ~ * . 3 J z - 1and K , c * c Knp1G K,, such that either (81) K,,-l = K,, or else (82) Kn-I c K,,, and Kn -clb(J) # 9. (83) In the case of (81), we have cl,(Jz) U clb(Jb,)= E since E -cla(Jt) = K,, = KnPlE clb(Jt), and the algorithm is done.

143) We now replace yo by y1 and return to (113). (144) Clearly, y1 has the property required by (113). (145) Jo relative to y1 has the property required by (116) of Jorelative to y". (146) Where El = E E: t(yl, j ) = c j } , J O G El is in keeping with requirement (115). Thus, (139) is a legitimate instruction. Matroid intersection 49 (147) In fact, by (83) and (138), the set K,,, which arose in the application of algorithm (75)-(90) to i@, and Jo, is contained in E l . To to a larger J in like structures which either augment f'to a larger J in F A n F t corresponding to y', or else lead as we have just described to a better dual solution y 2 and matroids ME.

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