By Stefan H. M. van Zwam

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**Additional resources for Combinatorial Mathematics [Lecture notes]**

**Example text**

Let ai := |Ai | and bi := |Bi | for all i ∈ [m]. Suppose Ai ∩ B j = if and only if i = j. Then m ai + bi −1 ai i=1 ≤ 1. 5: Let := {(A1 , B1 ), . . , (Am , Bm )} be a collection of pairs m of sets satisfying the conditions of the theorem. Let X := i=1 (Ai ∪ Bi ). We will prove the result by induction on |X |, the case |X | = 1 being easily verified. Suppose the claim holds for |X | = n−1, and assume |X | = n. For each x ∈ X , define x := {(Ai , Bi \ {x}) : (Ai , Bi ) ∈ , x ∈ Ai }. 2) holds for each x .

6 LEMMA. Define g(x 1 , . . , x m ; y) := x 1 ∆(x 1 + y, x 2 , . . , x m ) + x 2 ∆(x 1 , x 2 + y, . . , x m ) + · · · + x m ∆(x 1 , . . , x m + y). 8. WHERE TO GO FROM HERE ? 27 Then g(x 1 , . . , x m ; y) = x1 + · · · + x m + m 2 y ∆(x 1 , . . , x m ). Proof: Observe that g is a homogeneous polynomial of degree one more than the degree of ∆(x 1 , . . , x m ). Swapping x i and x j changes the sign of g (since ∆ can be seen to be alternating). Hence if we substitute x i = x j = x then g becomes 0.

Let A1 , . . , An be finite sets. An n-tuple (x 1 , . . , x n ) is a system of distinct representatives (SDR) if • x i ∈ Ai for i ∈ [n]; • x i = x j for i, j ∈ [n] with i = j. The question we wish to answer is: when does a set system A1 , . . , An have an SDR? Clearly each Ai needs to contain an element, and A1 ∪ · · · ∪ An needs to contain n elements. Write, for J ⊆ [n], A(J) := ∪i∈J Ai . A more general necessary condition, which is equally obvious, is Hall’s Condition: |A(J)| ≥ |J| for all J ⊆ N .