Families of arcs disconnected by finite sets I

✍ Scribed by M. Rochowski

• Publisher: John Wiley and Sons
• Year: 1975
• Tongue: English
• Weight: 140 KB
• Volume: 66
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19750660124

✦ Synopsis

By M. ROCHOWSKI of Katowice (Eingegangen am 5 . 12. 1973) 1. Introduction. I n this paper a generalization (theorem C,) of theorem Ci proved in [3] shall be formulated and as a consequence of it we prove MENOER'S n-Beinsatz (see [l], [2], [4]). The proof of theorem C, shall be published separately in a forthcoming paper of this journal. Let A , B denote two disjoint sets such that c a r d A , c a r d B z n + l , 1 i n < m , If F denote a family of sets, then by U F we denote the union of the sets of the family. Let F,(A, B ) denote a family of arcs which satisfies the following conditions: a,) every arc ab E F,(A, B ) has exactly the end points a, b in common with A U B, a n d a E A , b E B , b,J if an arc ub is contained in U F,,(A, B ) and has exactly the end points a, b in common with A U B, then ab E F,(A, B ) , c,) for every different n points a L , . . . , a,L E A resp. bl, . . . , b, E B there exist in F,L(A, B ) n disjoint arcs with the end points a , , . . . , a, resp. bl, . . . , b,, d,) among every n + 1 arcs of F,(A, B ) at least two have a point in common.

Then we have the following