New lower bounds for lattice sets avoiding subspheres and subspaces, including f_circ(n) ≥ 7n/12, via deletion-method counting of cyclic quadrilaterals and cospherical tuples.
Grid-drawings of graphs in three-dimensions
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abstract
Using probabilistic methods, we obtain grid-drawings of graphs without crossings with low volume and small aspect ratio. We show that every $D$-degenerate graph on $n$ vertices can be drawn in $[m]^3$ where $m^3 = O(D^2 n\log n)$. In particular, every graph of bounded maximum degree can be drawn in a grid with volume $O(n \log n)$.
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On subsets of lattice cubes avoiding affine and spherical degeneracies
New lower bounds for lattice sets avoiding subspheres and subspaces, including f_circ(n) ≥ 7n/12, via deletion-method counting of cyclic quadrilaterals and cospherical tuples.