Monte Carlo layer-ratio reconstruction via fixed-layer Markov chains produces the estimate M(10) ≈ 8.936 × 10^78 with uncertainty from cross-n scaling calibrated on known smaller values.
Counting maximal antichains and independent sets
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abstract
Answering several questions of Duffus, Frankl and R\"odl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the n-dimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.
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2026 1verdicts
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Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo
Monte Carlo layer-ratio reconstruction via fixed-layer Markov chains produces the estimate M(10) ≈ 8.936 × 10^78 with uncertainty from cross-n scaling calibrated on known smaller values.