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D B 0 2I m 1 jB hX i? Similarly, for each v 2 PV , we obtain ˇ ˝ ˛ˇˇ ˇ ? hX i? / D B 0 2I m 1 jB hX i? hXi? PV \ hXi? PV \ hvi? \ hXi? / n B 0 ˇ xB 0 : hX i? Let us show that ˇ ˇ ˝ ˛ˇˇ ˝ ˛ˇˇ ˇ ˇ ? ? PV \ hvi? \ hX i? / such that B 0 ( Li D PV \ hX i? PV \ hX i / n B ˇ uk vk D i ˇ ? 4 Examples of Posets with the Sperner Property 35 ˇ ˇ for i 2 Fq . PV \ hvi? \ hX i? / n hB 0 i. We will show that jLi j D ˇLj ˇ for all i; j , which implies the equation. PV \ hvi? \ PV \ hX i? , B 0 6 hvi? Hence there exists P x 2 B 0 such that hX i?

X; Y / j X i /. i C 1/. i C 1/. Since 2i < n, i C 1 Ä n i , which implies jS j Ä jT j. 2Œn / contains a full matching for 2i < n. Similar proof works for the case when 2i > n. 31, the Boolean lattice 2Œn has the Sperner property. P / is a poset. It is in fact a lattice as we will see below. Recall that a lattice is a poset in which any two elements have a least upper bound and a greatest lower bound. The least upper bound is sometimes called join and is denoted by x _ y and the greatest lower bound is called meet and is denoted by x ^ y.

N/. 58. n0 /. 59. x1i1 ; x2i2 ; : : : ; xkik // D i1 C i2 C C ik . Let n be a positive integer such that n D p1d1 p2d2 pkdk with distinct primes p1 ; p2 ; : : : ; pk . n/. 60. nr / N. It is a ranked finite poset. nj / for some j and z D p1i1 p2i2 pkik be the prime factorization of z. z/ is the “number of prime factors” that occur with counting multiplicity in the prime factorization of an integer in P . 4 Examples of Posets with the Sperner Property 21 jP0 j D 1; jP1 j ; jP2 j ; : : : ; jPs j is known as an O-sequence in commutative algebra.

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