Height Bounds for Ideals in Finite Posets

Definitions and mathematical statement

Let $P$ be a finite partially ordered set. An order ideal is a subset $A\subseteq P$ such that $x\in A$ and $y\leq x$ imply $y\in A$. Define \[ \max(A)=\{x\in A:\text{there is no }y\in A\text{ with }x<y\}. \] Here $|A|$ denotes cardinality. The expressions below require $A\neq\varnothing$, since the supplied record gives no convention for a maximum over the empty set.

Unresolved notation and scope. The supplied record does not define $h(x)$, its codomain, or any hypotheses connecting $h$ to $P$. In particular, the title does not establish that $h$ means ordinary poset height. Any restrictions on the intended class of posets are also missing. Consequently, the following is a conditional transcription of the supplied questions, not a fully specified conjecture about arbitrary finite posets.

Conjecture 12, conditional transcription. Once the intended definition of $h$ and the admissible posets have been supplied, is it true for every such $P$ and every nonempty order ideal $A\subseteq P$ that \[ \max_{x\in A}h(x)\geq |A|-|\max(A)|+1? \] The record also asks whether the deficit \[ |A|-\max_{x\in A}h(x) \] is bounded by a function of $|\max(A)|$ alone. Under a real-valued interpretation of $h$ and an upper-bound interpretation of "bounded," this asks whether there exists a function $f:\mathbb Z_{\geq1}\to\mathbb R$, independent of $P$ and $A$, such that for every admissible $P$ and every nonempty order ideal $A\subseteq P$, \[ |A|-\max_{x\in A}h(x)\leq f\left(|\max(A)|\right). \] These interpretations require confirmation from the missing context.

The original excerpt asserts that the first inequality is true and tight when $A=P$, mainly because of its reference [14]. This assertion is retained as a source qualification under the unspecified original hypotheses; it is not independently verified here. The surrounding excerpt also mentions Theorem 8, two possible strengthenings, and an old conjecture attributed to [7], without enough context to identify their relationship to these questions.

Short English statement

The supplied question asks whether the largest value of an undefined quantity $h$ on a nonempty order ideal is at least the ideal's size minus its number of maximal elements plus one. A weaker question asks for a uniform bound on the resulting deficit using only the number of maximal elements. Defining $h$ and recovering the intended hypotheses are necessary before either question is mathematically precise.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006544$, title Height Bounds for Ideals in Finite Posets, source code catalogue entry. The supplied excerpt labels the inequality "Conjecture 12." No proposer is identified; the proposed year is recorded as 2026.
• Supplied report citation: Combinatorics (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/1, pp. 5--79. https://doi.org/10.4171/owr/2026/1. This is an authoritative report citation supplied by the record; its contents and the missing definition of $h$ have not been independently checked here.
• The record labels the statement corrected_verified and says that the official report was checked and the notation repaired. These are supplied verification claims, not verification performed for this draft. The record labels the problem open.
• Supplied literature assessment, dated exactly 2026-08-22: a cylindric-shape nonnegative-coefficient conjecture is described as unresolved in general, with special fence and circular-fence cases established. This is supplied dated curation, not independently verified, and its subject does not match the ideal inequality in this record.
• Supplied literature source: Oberwolfach Report 2026/2. https://doi.org/10.4171/owr/2026/2.
• Supplied literature source: Kantarci Oguz--Ravichandran, Discrete Mathematics. https://doi.org/10.1016/j.disc.2022.113218.
• Supplied literature source: Skew shapes, Ehrhart positivity and beyond. https://arxiv.org/abs/2503.16403.
• The excerpt cites references [7] and [14] and mentions Theorem 8, but supplies neither their bibliographic identities nor their statements. No identification is inferred.