Abstract
AbstractA random graph model on a host graph is said to be 1‐independent if for every pair of vertex‐disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . For an infinite connected graph , the 1‐independent critical percolation probability is the infimum of the such that every 1‐independent random graph model on in which each edge is present with probability at least almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that tends to a limit in as , and they asked for the value of this limit. We make progress on a related problem by showing that In fact, we show that the equality above remains true if the sequence of complete graphs is replaced by a sequence of weakly pseudorandom graphs on vertices with average degree . We conjecture the answer to Balister and Bollobás's question is also .
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