By Stefan H. M. van Zwam

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Filled with the numbers 1 to n, each occurring once. . • . . such that each row and each column is increasing. 5(a) for an example. Young Tableaux play an important role in the theory of representations of the symmetric group. In this section we will content ourselves with counting the number f (n1 , . . , nm ) of tableaux of a specific shape λ = (n1 , . . , nm ). We start by finding a recursion. 2 LEMMA. f satisfies (i) f (n1 , . . , nm ) = 0 unless n1 ≥ n2 ≥ · · · ≥ nm ≥ 0; (ii) f (n1 , .

5. 3 THEOREM (Hales-Jewett). For all t, r ∈ there is a least dimension H J(t, r) such that for all n ≥ H J(t, r) and for all r-colorings of [t]n , there exists a monochromatic combinatorial line. The main issue in understanding the proof is getting a grip on the many indices that are floating around. We will frequently look at products of spaces, and write An1 +n2 = An1 × An2 . We will extend this notation to members of these spaces. For instance, if A = {1, 2, 3, 4}, and x, y ∈ A2 , say x = (1, 3) and y = (4, 2), then we write x × y ∈ A2 × A2 ; in particular x × y = (1, 3, 4, 2).

Case II. Suppose there is a nontrivial critical set. Pick J a minimum-size, nonempty critical set. By induction, {A j : j ∈ J} has an SDR X . Now define Ai := Ai \ A(J) for i ∈ J. For K ⊆ [n] \ J we find |A (K)| = |A(J ∪ K)| − |A(J)| ≥ |J ∪ K| − |A(J)| = |J ∪ K| − |J| = |K|, so (HC) holds for the Ai . By induction, there is an SDR Y for those sets. The union of X and Y then forms an SDR for the original problem. The name of the theorem derives from the following interpretation: men {1, . . , n} are trying to find a spouse; the set Ai denotes the eligible women for man i.