We prove that for$n$sufficiently large, all perfect 3-hash linear codes in$\mathbb {F}_{3}^{n}$must have dimension at most$\left ({\frac {1}{4}-\epsilon }\right)n$for some absolute constant$\epsilon > 0$.

The local properties problem of Erd\Hos and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants, by extending the color energy technique---a variant of the additive energy technique from additive combinatorics (color energy was originally introduced by the last two authors [C. Pohoata and A. Sheffer, Combinatorica, 39 (2019), pp. 705--714]). We generalize the concept of color energy to higher color energies and combine these with bounds on the extremal numbers of even cycles. Let $f(n,k,\ell)$ denote the minimum number of colors required to color the edges of $K_n$ such that every $k$ vertices span at least $\ell$ colors. It can be easily shown that $f(n,k,\binom{k}{2}-\lfloor \frac{k}{2}\rfloor +2)=\Theta(n^2)$. Erdös and Gyárfás asked what happens when $\ell=\binom{k}{2}-\lfloor k/2\rfloor +1$, one away from the easy case, and derived the bound $f\left(n,k,\ell\right) =\Omega(n^{4/3})$. Our technique significantly improves this to $f(n,k,\binom{k}{2} - \lfloor k/2\rfloor +1) = \Omega(n^{2- 8/k})$.