We survey results on the pebbling numbers of graphs as well as their historical connection with a number-theoretic 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.

In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the Kruskal-Katona Theorem and an analog of the Bollob as-Thomason 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.

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

Let F ⊂ 2 [n] be a 3-wise 2-intersecting 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

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension 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.

ABSTRACT. Let 1 ≤ t ≤ 7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A ∩ B ∩ C ∩ D | ≥ t holds for all A,B,C,D ∈ F. Then we have |F | ≤ �n−t � k k−t if | n − 1 2 | < ε holds for some ε> 0 and all n> n0(ε). We apply this result to get EKR type inequalities for “intersecting and union families ” and “intersecting Sperner families.” 1.

Abstract: We show that the Hilbert scheme, that parametrizes all ideals with the same Hilbert function over an exterior algebra, is connected. We give a new proof of Hartshorne’s Theorem that the classical Hilbert scheme is connected. More precisely: if Q is either a polynomial ring or an exterior algebra, we prove that every two strongly stable ideals in Q with the same Hilbert function are connected by a sequence of binomial Gröbner deformations. 1.

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.

ABSTRACT. What is the maximal size of k-uniform r-wise t-intersecting families? We show that this problem is essentially equivalent to determine the maximal weight of non-uniform r-wise t-intersecting families. Some EKR type examples and their applications are included. 1.

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 q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition. 1