Taylor's Condition

For a given Positive Integer $n$, does there exist a Weighted Tree with $n$ Vertices whose paths have weights 1, 2, ..., ${n\choose 2}$, where ${n\choose 2}$ is a Binomial Coefficient? Taylor showed that no such Tree can exist unless it is a Perfect Square or a Perfect Square plus 2. No such Trees are known except $n=2$, 3, 4, and 6.

See also Golomb Ruler, Perfect Difference Set

References

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1985.

Leech, J. ``Another Tree Labeling Problem.'' Amer. Math. Monthly 82, 923-925, 1975.

Taylor, H. ``Odd Path Sums in an Edge-Labeled Tree.'' Math. Mag. 50, 258-259, 1977.