By Gottfried Wilhelm Leibniz

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Let f 1; 2; : : : ; hk 1 g. Li;lj /i;j also has full rank. Hence the number of nonzero row vectors in L0 is at least n. k 1/ in Pk satisfying v 0 < v for some v 2 Pk0 1 D f vl1 ; vl2 ; : : : ; vln g Pk 1 is greater than or equal to n. x; y/ 2 Pk 1 Pk j x Ä y g contains a full matching. Similarly, we hk . 31. 50. h0 ; h1 ; : : : ; hs / is unimodal and symmetric. k/ for k < s=2. k/ has an inverse for each k, then P has the Sperner property. s Proof. 49, we have the lemma. k/ have full rank. 52. 51.

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? / and B 0 P 1 N D c x. Then xN satisfies i xN i vi D 1 and i xi vi D c for some c 2 Fq .

D dim hB 0 i? D n m for B; B 0 2 I m , we have n X 2 In m ˇ o n ˇ ? \ X 2 In ˇ hX i D hBi m ˇ ˝ ˛? o ˇ D; ˇ hX i D B 0 ? 0 ? hBi? / ˇ B 2 B m is Q-linearly independent. Hence f PB F j B 2 B g is a Q-linearly independent set consisting of jBj elements. I0 C I1 C I2 / are isomorphic. 85. I0 C I1 C I2 / is Gorenstein. This in particular proves that the sequence 0" # " # " # " # 1 n n n n A @ ; ; ;:::; 0 1 2 n q q q q is a Gorenstein vector. 71. I0 C I1 C I2 /. F /. k/ element of R. k/ of Q= AnnQ F of degree k.