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Given a finite poset P , we consider the largest size La(n, P ) of a family of subsets of [n] := {1, . . . , n} that contains no (weak) subposet P . This problem has been studied intensively in recent years, and it is conjectured that π(P ) := limn→∞ La(n, P )/ ( n b 2 c ) exists for general posets P , and, moreover, it is an integer. For k ≥ 2 let Dk(More)
Given a finite poset P , let La(n, P ) denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the nset, to give simpler new proofs of the known asymptotic behavior of La(n, P ), as n → ∞, when P is the(More)
The routing number rt(G) of a connected graph G is the minimum integer r so that every permutation of vertices can be routed in r steps by swapping the ends of disjoint edges. In this paper, we study the routing numbers of cycles, complete bipartite graphs, and hypercubes. We prove that rt(Cn) = n − 1 (for n ≥ 3) and for s ≥ t, rt(Ks,t) = 3s 2t + O(1). We(More)
Consider families of subsets of [n] := {1, 2, . . . , n} that do no contain a given poset P as a subposet. Let La(n, P) denote the largest size of such families and h(P) denote the height of P. The best known general upper bound for La(n, P) is ( 1 2 (|P| + h(P))− 1 ) ( n 2 ) , due to Bursi and Nagy (2012). This paper provides an improved upper bound 1 m+1(More)
Increasing attention is being paid to the study of families of subsets of an nset that contain no subposet P . Especially, we are interested in such families of maximum size given P and n. For certain P this problem is solved for general n, while for other P it is extremely challenging to find even an approximate solution for large n. It is conjectured that(More)
Given a finite poset P , we consider the largest size La(n, P ) of a family F of subsets of [n] := {1, . . . , n} that contains no subposet P . This continues the study of the asymptotic growth of La(n, P ); it has been conjectured that for all P , π(P ) := limn→∞ La(n, P )/ ( n b 2 c ) exists and equals a certain integer, e(P ). This is known to be true(More)
The vertex set of a Kneser graph KG(m,n) consists of all n-subsets of the set [m] = {0, 1, . . . ,m − 1}. Two vertices are defined to be adjacent if they are disjoint as subsets. A subset of [m] is called 2stable if 2 ≤ |a − b| ≤ m − 2 for any distinct elements a and b in that subset. The reduced Kneser graph KG2(m,n) is the subgraph of KG(m,n) induced by(More)
In this study, ZnO nanotubes (ZNTs) were prepared onto fluorine-doped tin oxide (FTO) glass and used as supports for MIPs arrays fabrication. Due to the imprinted cavities are always located at both inner and outer surface of ZNTs, these ZNTs supported MIPs arrays have good accessibility towards template and can be used as sensing materials for chemical(More)
Given a graded poset P , consider a chain decomposition C of P . If |C1| ≤ |C2| implies that the set of the ranks of elements in C1 is a subset of the ranks of elements in C2 for any chains C1, C2 ∈ C, then we say C is a nested chain decomposition (or nesting, for short) of P , and P is said to be nested. In 1970s, Griggs conjectured that every normalized(More)
Let m,n, and k be integers satisfying 0 < k 6 n < 2k 6 m. A family of sets F is called an (m,n, k)-intersecting family if ([n] k ) ⊆ F ⊆ ([m] k ) and any pair of members of F have nonempty intersection. Maximum (m, k, k)and (m, k + 1, k)-intersecting families are determined by the theorems of Erdős-KoRado and Hilton-Milner, respectively. We determine the(More)
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