Threshold conjecture for monotone properties of random geometric graphs

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Let Gd(n,p)G_d(n,p) be the random geometric graph model, and let F\mathcal{F} be a non-trivial monotone graph property, where monotonicity means monotonicity under adding edges. Random geometric graph threshold conjecture. Every non-trivial monotone property has a threshold in Gd(n,p)G_d(n,p). The analogous general theorem for G(n,p)G(n,p) is known, and special cases are known for d=1d=1; the general statement for random geometric graphs remains open.

References

Primary source

Will Perkins, “Searching for (sharp) thresholds in random structures: where are we now?”, arXiv:2401.01800 (2024).

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