Talagrands Combinatorics Conjecture

### What is the conjecture

Let $\delta \leq 1/2$ be a constant, $N$ an integer, and $P$ a product measure on $\{0,1\}^N$ where each coordinate has mass $\delta$ on $1$ and $1-\delta$ on $0$.

For a set $D \subset \{0,1\}^N$ and integer $q$, define:
$$D^{(q)} = \{x \in \{0,1\}^N : \forall x^1, \ldots, x^q \in D, \exists i \leq N, x_i = 1, x_i^1 = \cdots = x_i^q = 0\}$$

For a subset $I$ of coordinates, define:
$$B(I) = \{x \in \{0,1\}^N : \forall i \in I, x_i = 1\}$$

**Main Question:** Does there exist a constant $q$ independent of both $N$ and $\delta$ such that for each set $D$ satisfying $P(D) \geq 1 - 1/q$, one can find a family $\mathcal{G}$ of subsets of coordinates satisfying:
$$D^{(q)} \subset \bigcup_{I \in \mathcal{G}} B(I) \quad \text{and} \quad \sum_{I \in \mathcal{G}} \delta^{|I|} \leq 1/2$$

(This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.)

A proof is worth 1000$.

**Sources:**
- https://michel.talagrand.net/prizes/combinatorics/combinatorics.tex

### Prerequisites needed

**Formalizability Rating:** 2/5 (0 is best) (as of 2026-01-22)

Building blocks (from Mathlib):
- `Measure.prod` and `PMF` for product measures on finite types
- `Finset` for Boolean cubes and index sets
- Basic set operations, summation, and quantifiers

Missing pieces:
- Definition of $D^{(q)}$ (the non-covering characterization) as a custom predicate
- Connection between the abstract definition and explicit covering formulation with $B(I)$ sets

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-05
* ams-60

### Choose either option

- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

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