Every finite nearlattice with at least 83 times 2 to the (n minus 8) subnearlattices is planar, and for n at least 9 this bound is sharp.
One hundred twenty-seven subsemilattices and planarity
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abstract
Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$ subsemilattices.
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2019 1verdicts
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Planar semilattices and nearlattices with eighty-three subnearlattices
Every finite nearlattice with at least 83 times 2 to the (n minus 8) subnearlattices is planar, and for n at least 9 this bound is sharp.