Chain theorem conjecture for highly connected matroids and -spikes
Chain theorem conjecture for highly connected matroids and -spikes
Let be a positive integer. A matroid is -connected if it has the stated matroid connectivity property, and a -spike is a matroid with the -property. Consider matroids with no circuits or cocircuits of size .
Chain theorem conjecture. There exists a function such that if is a -connected matroid with no circuits or cocircuits of size and
then either there exists a -element set such that either or is -connected, or is a -spike.
This conjecture proposes that sufficiently large highly connected matroids either admit a connectivity-preserving deletion or contraction of a -element set, or belong to the structured exceptional class of -spikes. The source does not indicate that it has been resolved.
Sources & referencesView supporting material
Primary source
Nick Brettell, Rutger Campbell, Deborah Chun, Kevin Grace and Geoff Whittle, “On a generalisation of spikes”, arXiv:1804.06959 (2018).
Progress summary
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