Erdős Problem #192 — Arithmetic progressions in positive unit-coordinate lattice walks

About 47 years old · traced to 1979

Let $(a_i)_{i\geq1}$ be an infinite sequence in $\mathbb Z^d$ such that $a_{i+1}-a_i$ is always a positive coordinate unit vector. For which dimensions $d$ must the sequence contain three terms in arithmetic progression?

References

Primary source

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics, L'Enseignement Mathématique (2) 25 (1979), 325–344.

Additional references

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics, L'Enseignement Mathématique (2) 25 (1979), 325–344.

arithmetic-progressions combinatorics

InErdős Problems

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Open

No public discussion or published progress on this specific lattice-walk problem was found.

No public discussion or published progress was found for the specific threshold asserting that the property holds through dimension three and fails from dimension four onward.

Current status (as of June 2025): the problem appears open, with no recorded activity establishing or challenging the stated threshold.

No solutions have been posted yet.