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Statement

Fix integers

n \geq 2k

and

t \geq 1

. Two families

\mathcal{F}, \mathcal{G} \subseteq \binom{[n]}{k}

are \emph{cross-

t

-intersecting} if

|F \cap G| \geq t

for every

F \in \mathcal{F}

and every

G \in \mathcal{G}

. \textbf{Conjecture (Frankl-Tokushige):} For all

n \geq (t+1)(k-t+1)

,

|\mathcal{F}| \cdot |\mathcal{G}| \leq \binom{n-t}{k-t}^2

, with equality iff

\mathcal{F} = \mathcal{G} = \{F \in \binom{[n]}{k} : T \subseteq F\}

for some fixed

t

-set

T

. \textbf{Task:} Establish the conjecture in its full range of parameters

(n, k, t)

, including the regime currently open for small

t

.

Current frontier

For t = 1 (cross-intersecting), the bound binom(n-1, k-1)^2 is known when n ≥ 2k (Pyber 1986; Matsumoto-Tokushige 1989). For general t, the cross-t-intersecting conjecture is proved for every fixed t ≥ 14 outside a finite set of (n, k) pairs (recent work; the cited Tokushige conjecture refined in arXiv:2410.22792, 2024); small t and intermediate n remain open.

When this counts as solved

A full solve is a rigorous proof of the Frankl-Tokushige cross-

t

-intersecting conjecture for ALL

n \geq (t+1)(k-t+1)

and all

t \geq 1

, with the extremal-uniqueness statement. Explicit counterexample families

\mathcal{F}, \mathcal{G}

in the conjectured regime with

|\mathcal{F}| \cdot |\mathcal{G}|

strictly exceeding

\binom{n-t}{k-t}^2

also close it. Resolving any previously-open parameter slice counts as a breakthrough: e.g., the conjecture for a specific small

t \in \{2, 3, \ldots, 13\}

in its full range, settling the remaining finite

(n, k)

cases for some fixed

t \geq 14

, or a uniform stability theorem strictly stronger than the current literature. The slice has to be verifiably open. New shifting arguments that don't settle a new parameter range, or alternative proofs of already-established cases, don't count.

Classification

open-completion