Tight rainbow Turán number for paths
Determine the exact asymptotic, or tight, value of the largest function f(d) such that every properly edge-coloured graph with average degree at least d contains a rainbow path of length f(d), including whether the conjectured upper bound \(f(d)\leq\lceil d\rceil-1\) is sharp up to at most an additive constant.
References
In comparison, no tight bound is known for $f(d)$ despite considerable effort in this direction. Note that \Cref{lem:GreedyOnePath} gives an easy bound of $f(d) \geq d/4$. The current best known lower bound is $f(d) \geq 7d/18 - O(1)$ by Ergemlidze, Gy\"ori and Methuku, and the best upper bound is $f(d) \leq \lceil d \rceil - 1$ for $d > 2$ by Johnston and Rombach, coming from the coloured Cayley graph $\mathrm{Cay}(\mathbb F_2k, S)$ generated by a zero-sum set $S$ of size $\lceil d \rceil$. This upper bound is known to give the correct value of $f(d)$ for $2 < d \leq 6$ , and Halfpap has conjectured it to be tight for all $d$, up to possibly an additive constant.