For every fixed integer d≥ 1, let [n]^d=1,…,n^d, let M([n]^d) denote the maximum cardinality of a sum-free subset of [n]^d, and let N_d(n) be the number of subsets S⊆ [n]^d such that there are no x,y,z∈ S with x+y=z. Then, as n→∞, N_d(n)=2^M([n]^d)+o(n^d). Equivalently, the number of sum-free subsets of [n]^d is asymptotically governed, on the exponential scale, by the maximum size of a sum-free subset.

Elsholtz–Rackham conjecture on sum-free subsets of lattice cubes

The conjecture concerns the asymptotic number of sum-free subsets of lattice cubes. Elsholtz and Rackham resolved the two-dimensional case; the all-dimensional counting claim remains unsettled.

Known results

Dimension $2$ : density $3/5$ and the corresponding enumeration, Elsholtz–Rackham, 2017.
Dimensions $3$ and $4$ : extremal-size conjecture, Lepsveridze–Sun; the cited source gives no publication date.
Dimension $2$ : stability near the extremal stripe, supporting enumeration but not extending it to higher dimensions.

Version-2 all-dimensional extremal claim (date not stated)

A preprint claims that every dimension satisfies the conjectured maximum-size bound $|S|\leq c_d^*n^d+O(n^{d-1})$ , with extremizers given by hyperplane slices. This is a substantial claimed advance, but it addresses extremal cardinality rather than the number of sum-free subsets, and is neither refereed nor independently verified.

Current status (as of August 2026): The two-dimensional counting case is settled, while the counting conjecture in higher dimensions remains open; an unverified preprint claims only the related all-dimensional extremal-size result.

Elsholtz–Rackham conjecture on sum-free subsets of lattice cubes

Primary source

Additional references

Progress summary

Solutions 0