By Daniel I.A. Cohen

This vintage (1978) textbook is 30 years outdated, yet nonetheless very precious and relevent. It covers the elemental undergraduate path in combinatorial common sense, thought, and perform; no one has ever performed it larger. in addition to a transparent and easy-to-understand exposition, this ebook has the very best challenge units that i've got ever obvious. There are, I admit, a few extra complex books with contemporary effects. but when you really need to appreciate this topic, and while you are keen to paintings via countless numbers of good difficulties, i will warrantly you that Cohen is your guy!

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**Example text**

Thus, assume that fm (ac ) = fm (a) ≤ rdk . By (b) we can ﬁnd b0 , . . , b2k ∈ Bdk such that rdk−1 < fm (b0 ) < · · · < fm (b2k ). By the Pigeon Hole Principle we can ﬁnd an I ∈ [{0, . . , 2k}]k+1 and a b ∈ {a, ac } such that fm (b ∧ bi ) = fm (bi ) for all i ∈ I. Without loss of generality we can assume that I = {0, . . , k} and b ∧ bi = bi for all i ≤ k. Then fm (bc ∨ bi ) > rdk for all i ≤ k. Moreover, for every j < m there is at most one ij ≤ k for which fj (bc ∨ bij ) ≤ rdk , since for diﬀerent i, i ≤ k we have fj ((bc ∨ bi ) ∧ (bc ∨ bi )) = fj (bc ) > rdk .

1 CPAgame cube and disjoint coverings 33 Proof. Let f ∈ Fcube be such that f [Cω ] = P . 5, we can ﬁnd a subcube Q of P such that Q ⊂ P \ i<ω Pi . If P ∩ i<ω Pi is not meager in P , then there exists an i < ω such that P ∩ Pi has a nonempty interior in P . Thus, there exists a basic clopen set C in Cω , which is a perfect cube, such that f [C] ⊂ Pi . So, Q = f [C] is a desired subcube of P . 1. 2 used with D(P ) in place of P . For a singleton P ∈ Perf ∗ (X) we just put Q(P, P ) = P . Consider the following strategy S for Player II: S( Pη , Qη : η < ξ , Pξ ) = Q({Qη : η < ξ}, Pξ ).

For every x ∈ X deﬁne hx : [ω]2 → {0, 1} by putting for every n < m < ω hx (n, m) = 1 if and only if fn (x) ≤ fm (x). Since F is Ramsey and W generates F, we can ﬁnd a Wx ∈ W and an ix < 2 such that hx [[Wx ]2 ] = {ix }. Thus, the sequence Sx = fn (x) n∈Wx is monotone. It is increasing when ix = 1 and it is decreasing for ix = 0. i For W ∈ W and i < 2, let PW = {x ∈ X: Wx = W & ix = i}. Then i {PW : W ∈ W & i < 2} is a partition of X and for every W ∈ W and i < 2 i the sequence fn PW n∈W is monotone and pointwise convergent to some i function f : PW → R.