Hausdorff dimension of the largest increasing subset

Prove that, almost surely, the supremum of the Hausdorff dimensions of increasing subsets of the support of the Brownian separable permuton with parameter p equals α(p), and clarify the relationship between the limiting variable X(p) and the maximal α(p)-dimensional Hausdorff measure of such subsets.

Background

The paper relates longest increasing subsequences in sampled permutations to increasing subsets of the random support of the Brownian separable permuton. The conjecture proposes that the discrete growth exponent α(p) has a geometric interpretation as the maximal Hausdorff dimension of an increasing subset of that support.

The authors also conjecture a connection between X(p) and the maximal Hausdorff measure, potentially with a gauge correction, but do not establish either assertion.

References

Then, we conjecture that for all $p \in (0,1)$, almost surely,

\underset{A\in {I}_{p}{\sup} \; {\dim A } = \alpha(p),

where $\alpha(p)$ is defined as in \cref{thm:main_permutations}, and $\dim(\cdot)$ denotes the Hausdorff dimension.

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Section 1.3, paragraph “Hausdorff dimension of the largest increasing subset of the Brownian separable permuton”