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**Additional resources for An Introduction to Combinatorics and Graph Theory [Lecture notes]**

**Example text**

Vk , v1 }, where all of the vi are distinct. ) 8. Show that 8 < R(3, 4) ≤ 10. 9. Show that R(3, 4) = 9. 7 Sperner's Theorem The binomial coefficients count the subsets of a given set; the sets themselves are worth looking at. 1 Let [n] = {1, 2, 3, . . , n}. Then 2[n] denotes the set of all subsets of [n], and nk denotes the set of subsets of [n] of size k. 2 Sperner’s Theorem 35 Let n = 3. 3 A chain in 2[n] is a set of subsets of 2[n] that are linearly ordered by inclusion. An anti-chain in 2[n] is a set of subsets of 2[n] that are pairwise incomparable.

N − 1)! n+i−1 n+i−1 = (−1)i = (−1)i . i n−1 Thus ∞ ∞ n+i−1 −n i n+i−1 i (x + 1) = (−1) x = (−x)i . n−1 n−1 i=0 i=0 Now replacing x by −x gives −n (1 − x) ∞ = i=0 −n So (1 − x) is the generating function for 1, ∞ · 2, . . , ∞ · n} of size i. n+i−1 i x. 1 Newton’s Binomial Theorem 53 In many cases it is possible to directly construct the generating function whose coefficients solve a counting problem. 3 Find the number of solutions to x1 + x2 + x3 + x4 = 17, where 0 ≤ x1 ≤ 2, 0 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5, 2 ≤ x4 ≤ 6.

For some values i1 , i2 , . . , ik , x ∈ Aim , 1 ≤ m ≤ k, and x is not in the remaining sets Ai . Then x is counted zero times by any term involving an Ai with i ∈ / {i1 , i2 , . . , ik }, and is counted once, positively or negatively, by each term involving only Ai1 , Ai2 , . . , Aik . There are k terms of the form −|Aim |, which count x a total of −k times. 1 The Inclusion-Exclusion Formula 45 terms of the form |Ail ∩ Aim |, counting x a total of k2 times. Continuing in this way, we see that the final count for x on the right hand side is k k k − + · · · + (−1)k , 2 3 k 1−k+ or more compactly k k .

### An Introduction to Combinatorics and Graph Theory [Lecture notes] by David Guichard

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