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64
A Survey of Graph Pebbling
 Congr. Numer
, 1999
"... We survey results on the pebbling numbers of graphs as well as their historical connection with a numbertheoretic question of Erdös and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a gr ..."
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Cited by 40 (14 self)
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We survey results on the pebbling numbers of graphs as well as their historical connection with a numbertheoretic question of Erdös and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a graph equals its number of vertices, and second the pebbling threshold function for various natural graph sequences. Finally, we relate the question of the existence of pebbling thresholds to a strengthening of the normal property of posets, and show that the multiset lattice is not supernormal.
List Decoding of qary ReedMuller Codes
 IEEE Trans. Inform. Theory
, 2004
"... The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in ..."
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Cited by 23 (1 self)
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The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in [1] and [27]. The algorithm in [27] is an improvement of the algorithm in [1], it works for up to E < n(1 − √ 2u/q) errors but is applicable only to codes RMq(u, m) with u < q/2. In this paper, we will propose some deterministic listdecoding algorithms for qary ReedMuller codes. Viewing qary ReedMuller codes as codes from order domains, we present a listdecoding algorithm for qary ReedMuller codes, which is a straightforward generalization of the listdecoding algorithm of ReedSolomon codes in [9]. The algorithm works for up to n(1 − m+1 √ u/q) m − 1 errors, and it is applicable to codes RMq(u, m) with u < q. The algorithm can be implemented to run in time polynomial in the length of the codes. Following [12], we show that qary ReedMuller codes are subfield subcodes of ReedSolomon codes. We then present a second listdecoding algorithm for qary ReedMuller codes. This algorithm works for codes with any rates, and achieves an errorcorrection bound n(1 − √ (n − d)/n) − 1. So the second algorithm achieves a better errorcorrection bound than the algorithm in [27], since when u is small, n(1 − √ (n − d)/n) = n(1 − √ u/q). The implementation of the second algorithm requires O(n) field operations in Fq and O(n3) field operations in Fqm under some assumption. Also, we prove that qary ReedMuller codes can be described as onepoint AG codes. And using the algorithm of AG codes in [9], we give a third listdecoding
Face vectors of flag complexes
"... Abstract. A conjecture of Kalai and Eckhoff that the face vector of an arbitrary flag complex is also the face vector of some particular balanced complex is verified. 1. ..."
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Cited by 22 (3 self)
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Abstract. A conjecture of Kalai and Eckhoff that the face vector of an arbitrary flag complex is also the face vector of some particular balanced complex is verified. 1.
Thresholds for Families of Multisets, With an Application to Graph Pebbling
, 2000
"... In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the KruskalKatona Theorem and an analog of the Bollob asThomason threshold result. As a corollary we obtain the existence of pebbling thresholds for arbitrary graph sequences. I ..."
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Cited by 20 (15 self)
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In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the KruskalKatona Theorem and an analog of the Bollob asThomason threshold result. As a corollary we obtain the existence of pebbling thresholds for arbitrary graph sequences. In addition, we improve both the lower and upper bounds for the `random pebbling' threshold of the sequence of paths.
Triangulated manifolds with few vertices: Combinatorial manifolds
, 2005
"... Let M be a simplicial manifold with n vertices. We call M centrally symmetric if it is invariant under an involution I of its vertex set which fixes no face of M. Obviously, the number of vertices of a centrally symmetric (triangulated) manifold is even, n = 2k, and, without loss of generality, we m ..."
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Cited by 18 (1 self)
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Let M be a simplicial manifold with n vertices. We call M centrally symmetric if it is invariant under an involution I of its vertex set which fixes no face of M. Obviously, the number of vertices of a centrally symmetric (triangulated) manifold is even, n = 2k, and, without loss of generality, we may assume that the involution is presented by the permutation I = (1 k+1)(2 k+2) · · ·(k 2k). The boundary complex ∂C ∆ k of the kdimensional crosspolytope C ∆ k is clearly centrally symmetric with respect to the standard antipodal action, and a subset F ⊆ {1, 2,...,2k} is a face of ∂C ∆ k if and only if it does not contain any minimal nonface {i, k + i} for 1 ≤ i ≤ k. Hence, every centrally symmetric manifold with 2k vertices appears as a subcomplex of the boundary complex of the kdimensional crosspolytope. Free Z2actions on spheres are at the heart of the BorsukUlam theorem, which has an abundance of applications in topology, combinatorics, functional analysis, and other areas of mathematics (see the surveys of Steinlein [50],
Generalized Hamming weights of qary ReedMuller codes
 IEEE Trans. Inform. Theory
, 1998
"... Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the qary ReedMuller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain c ..."
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Cited by 17 (1 self)
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Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the qary ReedMuller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition. 1
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 16 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Weighted 3Wise 2Intersecting Families
 J. COMBIN. THEORY (A
, 2002
"... Let n and r be positive integers. Suppose that a family satisfies 2 for all F 1 , F 2 , F 3 . We prove that if w < 0.5018 then F#F w F  (1 . 1 ..."
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Cited by 13 (12 self)
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Let n and r be positive integers. Suppose that a family satisfies 2 for all F 1 , F 2 , F 3 . We prove that if w < 0.5018 then F#F w F  (1 . 1
Random walks and multiply intersecting families
 J. Combin. Theory (A
, 2005
"... Let F ⊂ 2 [n] be a 3wise 2intersecting Sperner family. It is proved that n−2 if n even, (n−2)/2 F  ≤ � � n−2 + 2 if n odd (n−1)/2 holds for n ≥ n0. The unique extremal configuration is determined as well. 1 ..."
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Cited by 10 (10 self)
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Let F ⊂ 2 [n] be a 3wise 2intersecting Sperner family. It is proved that n−2 if n even, (n−2)/2 F  ≤ � � n−2 + 2 if n odd (n−1)/2 holds for n ≥ n0. The unique extremal configuration is determined as well. 1
Lexifying ideals
 MATH. RES. LETTERS
"... This paper is on monomial quotients of polynomial rings over which Hilbert functions are attained by lexicographic ideals. ..."
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Cited by 10 (2 self)
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This paper is on monomial quotients of polynomial rings over which Hilbert functions are attained by lexicographic ideals.