Growth constants of Lipschitz functions on hypercubes
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For the Lipschitz-function growth constant c(Q_d) on the d-dimensional hypercube, determine its asymptotic behaviour as d grows. The stated upper-bound target is log c(Q_d)≤(3/4+o(1))(log d)/d.
Why it matters and what counts as progress
Why it matters: It quantifies the growth of constrained functions on increasingly high-dimensional cubes.
What counts as progress: A sharper asymptotic upper or lower bound for c(Q_d), particularly the coefficient of (log d)/d in its logarithm.
How the problem developed
- 14 Jan 2024 · Definition and bounds [1]
Samuel Korsky, Tahsin Saffat, and Dhroova Aiylam Introduced a growth constant for integer-valued Lipschitz functions on finite graphs and proved bounds for grids and sparse random graphs.
- 25 May 2026 · Bounds [2]
Samuel Korsky Established hypercube bounds π²/(6d)+o(1/d) ≤ log c(Q_d) ≤ (3/4+o(1))log(d)/d.
AI contribution
Lipschitz Functions on Sparse Graphs II
For the Lipschitz-function growth constant c(Q_d) of the d-dimensional hypercube, log c(Q_d)≤(3/4+o(1))(log d)/d. The argument compares graph homomorphisms with K_(d,d).
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