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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 log 2 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)

- Karen Gunderson, Michał Przykucki
- 2014

Bootstrap percolation is a cellular automaton modelling the spread of an 'infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for r-neighbour bootstrap percolation on Galton–Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollobás, Gunderson,… (More)

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