Chain theorem conjecture for highly connected matroids and tt-spikes

Let tt be a positive integer. A matroid is (2t1)(2t-1)-connected if it has the stated matroid connectivity property, and a tt-spike is a matroid with the (t,2t)(t,2t)-property. Consider matroids with no circuits or cocircuits of size 2t12t-1.

Chain theorem conjecture. There exists a function f(t)f(t) such that if MM is a (2t1)(2t-1)-connected matroid with no circuits or cocircuits of size 2t12t-1 and

E(M)f(t),|E(M)| \ge f(t),

then either there exists a tt-element set XE(M)X \subseteq E(M) such that either M/XM/X or M\XM \backslash X is (t+1)(t+1)-connected, or MM is a tt-spike.

This conjecture proposes that sufficiently large highly connected matroids either admit a connectivity-preserving deletion or contraction of a tt-element set, or belong to the structured exceptional class of tt-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).

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