By David Guichard

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A cycle is a sequence of edges {v1 , v2 }, {v2 , v3 }, . . , {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.

We know that the number of solutions in nonnegative integers is 7+3−1 = 92 , so this is an overcount, since we count solutions that 3−1 do not meet the upper bound restrictions. For example, this includes some solutions with x1 ≥ 3; how many of these are there? This is a problem we can solve: it is the number of solutions to x1 + x2 + x3 = 7 with 3 ≤ x1 , 0 ≤ x2 , 0 ≤ x3 . This is the same as the number of non-negative solutions of y1 + y2 + y3 = 7 − 3 = 4, or 4+3−1 = 62 . Thus, 3−1 9 6 2 − 2 corrects this overcount.

N}. ∞ Bn · Let f (x) = n=0 xn , and note that n! ∞ ∞ ∞ xn−1 xn f (x) = Bn = Bn+1 = (n − 1)! n! n=1 n=0 n=0 n k=0 n Bn−k k xn , n! 4. Now it is possible to write this as a product of two infinite series: ∞ Bn · f (x) = n=0 xn n! ∞ an xn = f (x)g(x). n=0 Find an expression for an that makes this true, which will tell you what g(x) is, then solve the differential equation for f (x), the exponential generating function for the Bell numbers. 4, the first few Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437.