For a matroid M, a circuit-cocircuit intersection (CCI) is a subset of its ground set that is the intersection of a circuit and a cocircuit. Oxley's circuit-cocircuit intersection conjecture. A matroid with a CCI of size k≥4 has a CCI of size k-2. Oxley implicitly observed the conjecture for k=6, and Kingan and Lemos proved it for regular matroids. The paper establishes it for every CCI of size k≤7; the general case remains open.

Oxley's circuit-cocircuit intersection conjecture

About 5 years old · traced to 2021

For a matroid $M$ , a circuit-cocircuit intersection (CCI) is a subset of its ground set that is the intersection of a circuit and a cocircuit. Oxley's circuit-cocircuit intersection conjecture. A matroid with a CCI of size $k\ge4$ has a CCI of size $k-2$ . Oxley implicitly observed the conjecture for $k=6$ , and Kingan and Lemos proved it for regular matroids. The paper establishes it for every CCI of size $k\le7$ ; the general case remains open.

References

Primary source

Jaeho Shin, “The Circuit-Cocircuit Intersection Conjecture for Intersection Size 7”, arXiv:2308.02859 (2023).

Additional references

2 papers in this index state this conjecture (2021–2023). The statement above is taken from the most recent of them; the others are arXiv:2111.02605.

circuit-cocircuit-intersections matroid-theory

Posted 1 month ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

A new unrefereed paper claims a counterexample that would disprove the conjecture, while all previously established cases remain limited to small intersections.

Oxley conjectured that a matroid having a circuit–cocircuit intersection of size $k \ge 4$ also has one of size $k-2$ . The conjecture was implicitly observed by Oxley in 1984, formulated in 1992, and remains unsettled in general.

Known results

Oxley, 1984: the case $k=6$ .
Kingan and Lemos, 2006: the conjecture for regular matroids.
Shin, 2023: the conjecture for arbitrary matroids when $k \le 7$ .

October 2026 claimed counterexample

An October 5, 2026 report describes Houshan Fu’s unrefereed preprint, which proposes a self-dual binary matroid whose intersection sizes are only $0,2,4,6,8,12$ . If correct, the size- $12$ intersection has no size- $10$ counterpart, disproving Oxley’s conjecture; the construction has not been independently verified.

Current status (as of October 2026): Cases through $k=7$ are established, while a preprint claims a counterexample for the general conjecture but remains unverified.

Sources

No solutions have been posted yet.