By L. Mirsky
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Extra info for An account of some aspects of combinatorial mathematics
The pair (S, 9)is called a topological space; we say that it is finite when S is finite. ) The members of 9 are called the open sets of the topological space (S, 9). A particularly simple topological space may be constructed as follows. Let S be any (non-empty) set, and let 9 denote its power set. Then (S, 9) is plainly a topological space: we say that we have formed it by endowing S with the discrete topology. The space (S, 9) is called a discrete topological space. Let S be a topological space.
State the deltoid version of this result. + 2. 4. 3. 1. 4. Let (X, A, Y ) be a finite deltoid. Show that X (and so also Y ) possesses an admissible subset of cardinal r if and only if (X’ x Y ’ )n A # 0 whenever X’ E X, Y’ E Y , and IX’I lY’l > 1x1 + IYI - r. 5. 1. 3 Further extensions of Hall’s theorem By continuing to invoke the device of elementary constructions, we shall obtain a series of further results each of which yields Hall’s theorem as a special case. 1. Let 4I = (A,, .. , A,) be a ,family of subsets of E, and let p , , ..
Later, we shall see how to supplement this method with more powerful ideas, above all those based o n the notion of abstract independence. I n the present chapter, however, and also a t various stages of the subsequent developmcnt, we shall exploit to the full the potentialities of elementary const r uc t i o n s. 2 Transversal index Not every family of sets possesses a transversal. In the present section we shall investigate the problem of the maximum size of partial transversals. , A,,) is a finite family of sets, we shall understand by its trarisversul index, denoted by t*, the maximum cardinal of PTs of 91.
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