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Open math problems for undergraduates — 15 you can actually work on

Unsolved problems that an undergraduate can understand, explore by computer and make checkable progress on — Ramsey numbers, cap sets, Golomb rulers, sorting networks, Erdős problems and more — with what counts as a result for each.

Updated 2026-10-03 · CC BY 4.0

Most famous open problems — the Riemann hypothesis, the twin prime conjecture — are famous because they are out of reach. They make poor projects. A good undergraduate problem is one where the statement fits in a sentence, small cases can be explored by computer, and a partial result is still a real result: a better construction, a new verified case, a formalised lemma, a documented dead end.

The problems below are all open as of October 2026 and all have that shape. For each we give the current status and what would count as progress. Each links to a page on Cairn Commons with sources, the exact verification rules, and earlier attempts.

Construction problems: beat a record

In these, a better example is checked by a program in seconds. You do not need to prove anything deep — you need a good search.

1. The Ramsey number R(4,6). Find a graph on 36 vertices with no 4-clique and no independent set of size 6. Known: 36 ≤ R(4,6) ≤ 40. A single such graph is a new lower bound. Background: the R(4,6) guide and the table of small Ramsey numbers.

2. Cap sets in dimension 7. A cap set in F_3^n has no three points on a line. The largest in dimension 7 has between 236 and 291 points. Any cap of 237 points is new.

3. The Schur number S(6). Colour 1, …, N with six colours so that no colour class contains x, y and x + y. The best known colouring reaches N = 536 (2000). A longer one is a new bound. S(5) = 160 was found by a SAT solver in 2017, so the tools are within reach.

4. Snake-in-the-box. Find long induced paths in the n-dimensional hypercube. Optimal values are known only up to n = 8; for n = 9–13 the records are 191, 379, 746, 1476 and 2924.

5. Packing circles in a square. Place n equal circles in a unit square as large as possible. Optimal packings are proved only for n ≤ 33 and n = 36; best known packings for larger n are tabulated, and improvements still appear.

6. The Erdős minimum overlap problem. A constant known to lie between 0.379005 and 0.380868. The upper bound comes from an explicit step function; the record changed in 2025 and again in 2026, partly through AI-assisted search.

7. Covering designs. Find small families of k-subsets that cover every t-subset of a v-set. The La Jolla Covering Repository keeps thousands of records, many of which can be improved by local search.

Verification frontiers: push a computation further

Here the question is settled for small cases by computer, and the next case needs a better algorithm, not a bigger machine.

8. Optimal Golomb rulers. A ruler whose marks have all pairwise distances distinct. Optimality is proved for up to 28 marks (distributed.net, 2022), and nobody is currently searching 29 marks. Better pruning ideas are the way in.

9. Costas arrays of order 32. Permutation matrices with all displacement vectors distinct. Every order up to 29 has been enumerated, and the known constructions give arrays for most orders — but none is known for orders 32 and 33. Finding one, or proving there are none, would settle a decades-old question.

10. Sorting networks. The minimum number of comparators to sort n inputs is known for n ≤ 12. For 13 inputs the best network uses 45 comparators; is there one with 44?

11. Small van der Waerden numbers. Only seven non-trivial values are known, the latest W(3,4) = 293 from 2012. New lower-bound colourings are checkable; new exact values need SAT solving at scale.

Number theory with a computer

12. The Erdős–Straus conjecture. Is 4/n = 1/x + 1/y + 1/z solvable in positive integers for every n ≥ 2? Verified for all n ≤ 10^18 (2025). Most residue classes are covered by explicit identities; studying the ones that are not is a classic exercise that leads straight to the frontier. More problems like it are on the unit fractions topic page.

13. Erdős problems with a finite check. Many Erdős problems are marked "falsifiable" (a computation would refute them if false) or "decidable" (reduced to a finite computation). Each comes with a Lean statement, so a formal proof of a special case is also a contribution. See how to work on an Erdős problem.

Proof-oriented problems

14. The lonely runner conjecture. Proved for up to 13 runners by computer-assisted proofs in 2025–2026. Re-verifying a case with your own code, or formalising a reduction lemma in Lean, is a concrete project. Status: the lonely runner in 2026.

15. The chromatic number of the plane. How many colours are needed so that no two points at distance 1 share a colour? The answer is 5, 6 or 7. Since 2018 the race has been to find smaller unit-distance graphs that need 5 colours; the smallest known has 509 vertices. A smaller one is a checkable result.

How to make a project out of one

  1. Read the problem page and its sources. Know the current record and who holds it.
  2. Reproduce the record first. Writing your own checker for the known best example teaches you the problem and gives you a tool you will need anyway.
  3. Pick a narrow goal: one more vertex, one fewer comparator, one more verified case — or a careful negative result about why an approach fails.
  4. Use AI where it is strong: writing search code, explaining a paper, finding a mistake in your argument. Check everything it tells you about the literature.
  5. Get it checked. On Cairn Commons constructions are verified automatically, computations are re-run, and arguments are reviewed. Your name is on the result, under CC BY 4.0, and it counts even if it is "only" a documented dead end.

If you want something matched to your level without reading fifteen pages, let the site pick a task for you. Or browse problems by topic or collection.

Problems mentioned in this guide

  • Cap sets in dimension 7 — Find a cap set in F_3^7 larger than the best known construction, or prove a better upper bound.
  • Packing equal circles in a unit square — For each n, find the largest radius r such that n non-overlapping circles of radius r fit in a unit square. Optimality is proven only for small n. For larger n, improve the best known packings or prove new cases optimal.
  • Costas arrays of order 32 and 33 — Find a Costas array of order 32 or 33, the smallest orders for which none is known, or extend the complete enumeration of Costas arrays beyond order 29.
  • Smaller covering designs — Improve upper bounds C(v,k,t) for covering designs listed in the La Jolla Covering Repository.
  • Erdős minimum overlap problem — Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
  • The Erdős–Straus conjecture — Prove that 4/n = 1/x + 1/y + 1/z has a solution in positive integers for every n ≥ 2. It has been verified for all n ≤ 10^18, and all n outside a few residue classes are covered by explicit identities.
  • Optimal Golomb rulers — Find the shortest Golomb ruler (all pairwise mark differences distinct) with n marks. Optimality is proven up to 28 marks (length 585, distributed.net, 2022); 29 marks is the first open case, and shorter rulers for larger n would beat long-standing constructions.
  • The chromatic number of the plane (Hadwiger–Nelson problem) — Determine how many colours are needed so that no two points of the plane at distance exactly 1 share a colour. The answer is known to be 5, 6 or 7; a concrete sub-goal is a smaller 5-chromatic unit distance graph than the 509-vertex record.
  • The lonely runner conjecture — For k+1 runners with distinct constant speeds on a unit circular track, each runner is at some time at distance at least 1/(k+1) from all others. Computer-assisted proofs now cover up to 13 runners; the general case is open.
  • The Ramsey number R(4,6) — Narrow the gap 36 ≤ R(4,6) ≤ 40. A 2-colouring of K_36 with no red K_4 and no blue K_6 would raise the lower bound; lowering the upper bound needs reproducible exhaustive computation.
  • The sixth Schur number S(6) — Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.
  • Snake-in-the-box — longest induced paths in hypercubes — Find the longest induced path (snake) in the n-dimensional hypercube Q_n. Optimal lengths are known only up to n = 8 (98); for n = 9–13 new records were set in 2026 and further improvements are open.
  • Smallest sorting networks for 13+ inputs — Find sorting networks with fewer comparators than the best known for n ≥ 13 inputs, or prove optimality.
  • Small van der Waerden numbers — Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.

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How to work on an Erdős problem with ChatGPT (or any chatbot)Using Claude for math research — a practical workflowUsing ChatGPT for math research — what works and how to check itRemote MCP server with OAuth for Claude — a worked exampleHow to add a remote MCP server to ChatGPT (developer mode)Small Ramsey numbers — table of known values and bounds (2026)