Karen Gunderson

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An m × n matrix A with column supports {Si} is k-separable if the disjunctions ⋃ i∈K Si are all distinct over all sets K of cardinality k. While a simple counting bound shows that m > k log2 n/k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker(More)
Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number r, the r-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: ‘infected’ or ‘healthy’. In consecutive rounds, each healthy vertex with at least r infected(More)
For r ≥ 2, an r-uniform hypergraph is called a friendship r-hypergraph if every set R of r vertices has a unique ‘friend’ – that is, there exists a unique vertex x / ∈ R with the property that for each subset A ⊆ R of size r − 1, the set A ∪ {x} is a hyperedge. We show that for r ≥ 3, the number of hyperedges in a friendship r-hypergraph is at least r+1 r ((More)
Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a “small” graph H and a “large” graph G = G0 ⊆ Kn, in consecutive steps we obtain Gt+1 from Gt by adding to it all new edges e such that Gt ∪ e contains a new copy of H. We say that G percolates if for some t ≥ 0, we have Gt = Kn. For H = Kr , the(More)
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