Symmetric Intersecting Families in Discrete Product Spaces

Definitions and mathematical statement

Let $k\geq 3$ and $n\geq 1$ be integers, and write $[k]=\{1,\ldots,k\}$. A family $A\subseteq[k]^n$ is intersecting if \[ \forall x,y\in A\quad \exists i\in\{1,\ldots,n\}\quad x_i=y_i. \] For a precise coordinate-permutation interpretation of symmetry, let $S_n$ act on $[k]^n$ by $(\sigma\cdot x)_i=x_{\sigma^{-1}(i)}$, and define \[ G_A=\{\sigma\in S_n:\sigma\cdot A=A\}. \] Say that $A$ is coordinate-transitive if \[ \forall i,j\in\{1,\ldots,n\}\quad \exists\sigma\in G_A\quad \sigma(i)=j. \] Under this interpretation, define \[ M_k(n)=\max\left\{|A|:A\subseteq[k]^n\text{ is intersecting and coordinate-transitive}\right\}. \] The explicit extremal question is to determine sharp upper bounds for $M_k(n)$, including its asymptotic growth as $n\to\infty$ for each fixed integer $k\geq3$.

The broader source question asks for bounds after excluding highly asymmetric, single-coordinate extremal examples. A single-coordinate family has the form \[ D_{i,a}=\{x\in[k]^n:x_i=a\},\qquad i\in\{1,\ldots,n\},\quad a\in[k]. \] The source does not specify whether exclusion means forbidding equality with such a family, forbidding containment in one, or imposing a quantitative condition on resemblance to one. Thus this broader part does not yet define a unique extremal problem.

The definition of $G_A$ above specifies one interpretation of the transitive subquestion. The source does not state whether its automorphisms are restricted to coordinate permutations or may also permute alphabet symbols within coordinates; the latter convention could yield a different admissible class. The source also leaves the range of $n$ implicit. The convention $n\geq1$ makes the definitions explicit, but at $n=1$ coordinate transitivity is automatic and does not exclude single-coordinate families. These qualifications require mathematical review.

Short English statement

An intersecting family consists of words such that every pair agrees in some coordinate. The explicit subquestion asks how large such a family can be when its coordinate symmetries act transitively. The intended automorphism convention and the broader rule excluding single-coordinate examples remain unspecified. The supplied dated curation labels the problem partially solved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004357$, title Symmetric Intersecting Families in Discrete Product Spaces, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is $2020$; no individual proposer is identified.
• Supplied report citation: Combinatorics (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/1, pp. 6--89. https://doi.org/10.4171/owr/2020/1. The supplied background describes these reports as documenting mathematical workshops and open-problem discussions.
• Supplied literature reference: Eberhard--Kahn--Narayanan--Spirkl, https://arxiv.org/abs/1909.11578. No title, theorem number, or precise correspondence between its symmetry convention and this record is supplied.
• Literature assessment dated 2026-08-22: the supplied curation states that, for fixed alphabet size at least three, symmetric intersecting families have size $o(k^n)$, while the broader extremal classification after excluding one-coordinate examples remains incomplete. The supplied status is partially_solved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This is supplied editorial metadata, not independent mathematical verification of this draft.