Bounded additive gap above the logarithmic leaf bound

Determine whether the maximum isometric ell-infinity embedding dimension D(t) of a finite tree with t leaves satisfies D(t) leq eil log_2 teil+1 for every tgeq2.

Background

For a finite tree with t leaves, the paper defines D(t) as the maximum least dimension required for an isometric embedding into ell-infinity. The universal leaf-count lower bound is eil log_2 teil. The paper proves that this lower bound is attained through 31 leaves but fails at 32 leaves, where D(32)=6 rather than 5. The resulting extremal question asks whether the failure can always be bounded by an additive one coordinate.

References

This leads to the following extremal question. Is $D(t)\le\lceil\log_2t\rceil+1$ for every $t\ge2$?

— A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding  (2608.16288 - Chalmers, 17 Aug 2026) in Section 5, Consequences, Problem (5.1)

More generally, it is unknown whether $\Delta(t)$ is bounded, or whether $D(t)=(1+o(1))\log_2t$.

— A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding  (2608.16288 - Chalmers, 17 Aug 2026) in Section 5, Consequences, immediately after Problem (5.1)

More generally, it is unknown whether $\Delta(t)$ is bounded, or whether $D(t)=(1+o(1))\log_2t$.

— A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding  (2608.16288 - Chalmers, 17 Aug 2026) in Section 5, Consequences, immediately after Problem (5.1)