Determine the even-parameter rainbow Turán numbers outside the established cases

Determine the asymptotic value of \(\mathrm{ex}^*(n,B_{t,3})\) for even integers \(t\) not already covered by the known exact cases, including its position within the bounds established for even \(t\).

Background

For even tt, the paper proves a general upper and lower bound and improves the upper bound when t≡0(mod4)t\equiv 0\pmod 4. Exact asymptotics are known when t=2s−2t=2^s-2, while additional cases are settled by constructions for powers of two and one less than powers of three. The authors explicitly identify the remaining location of the extremal value within the available range as unresolved.

References

If t$ is not of the form $2s - 2$, it is not clear where $\mathrm{ex*}(n,B_{t,3})$ should fall within the range given by Theorem~\ref{even t upper bound}.

— Rainbow Turán numbers for short brooms  (2502.16057 - Byrne et al., 22 Feb 2025) in Section 3.2, subsection “Upper bound for \(t \equiv 0 \pmod 4\)”