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A random graph model on a host graph H is said to be 1-independent if forevery pair of vertex-disjoint subsets A,B of E(H), the state of edges (absentor present) in A is independent of the state of edges in B. For an infiniteconnected graph H, the 1-independent critical percolation probability$p_{1,c}(H)$ is the infimum of the p in [0,1] such that every 1-independentrandom graph model on H in which each edge is present with probability at leastp almost surely contains an infinite connected component.  Balister and Bollob\'as observed in 2012 that $p_{1,c}(\mathbb{Z}^d)$ isnonincreasing and tends to a limit in [1/2, 1] as d tends to infinity. Theyasked for the value of this limit. We make progress towards this question byshowing that \[\lim_{n\rightarrow \infty}p_{1,c}(\mathbb{Z}^2\timesK_n)=4-2\sqrt{3}=0.5358\ldots \ .\] In fact, we show that the equality aboveremains true if the sequence of complete graphs $K_n$ is replaced by a sequenceof weakly pseudorandom graphs on n vertices with average degree $\omega(\logn)$. We conjecture that the equality also remains true if $K_n$ is replacedinstead by the n-dimensional hypercube $Q_n$. This latter conjecture wouldimply the answer to Balister and Bollob\'as's question is $4-2\sqrt{3}$.  Using our results, we are also able to resolve a problem of Day, Hancock andthe first author on the emergence of long paths in 1-independent random graphmodels on $\mathbb{Z}\times K_n$. Finally, we prove some results on componentevolution in 1-independent random graphs, and discuss a number of open problemsarising from our work that may pave the way for further progress on thequestion of Balister and Bollob\'as.

$1$-independent percolation on $\mathbb{Z}^2 \times K_n$