On the Gap of Finite Posets

@misc{haqi2026finite,
  title = {{On the Gap of Finite Posets}},
  author = {Alireza Haqi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.12678},
  eprint = {2608.12678},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.12678},
}

TY  - UNPB
TI  - On the Gap of Finite Posets
AU  - Alireza Haqi
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.12678
UR  - https://arxiv.org/abs/2608.12678
AB  - Let P be a finite nonempty poset with n elements, let f:P→{1,…,n} be a uniformly random order-preserving bijection, and put h_P(x)=𝔼[f(x)]. Aires and Kahn (2025) introduced gap(P) as the largest difference between consecutive values in the ordered list consisting of $0$, n+1, and all the expected ranks h_P(x). Write w(P) for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound gap(P)≤ 2 w(P)−1. Second, for every L>0 we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least L, with $0$ and |P|+1 added as endpoints. Finally, for every r∈ℕ, we construct a poset P_r for which the relative order induced on every nonempty selected set X has base-two entropy below 3|X|, while gap(P_r)≥(3/2)^r. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.
ER  -