Closing the triangle-factor budget gap

Determine whether triangle factors admit a successful strategy with exposure time $t=\Theta(n^{4/3}\log^{1/3}n)$ and budget $b=o(n^{4/3}\log^{1/3}n)$.

Background

For partial F-factors, the paper establishes matching time-budget bounds up to constants, but for complete strictly 1-balanced F-factors it leaves a polylogarithmic gap. Triangle factors are presented as the simplest unresolved instance of whether the logarithmic factor can be removed near the expected hitting time.

References

Does there exist a successful $(t,b)$-strategy for triangle factors with $t=\Theta(n{4/3}\log{1/3}n)$ and $ b = o(n{4/3} \log{1/3}n)$?

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Problem 7.1, Section 7 “Final remarks and open problems”