The wasteful partial linkage hindrance conjecture for finitary matroids

Let MM and NN be finitary matroids on a common ground set. For a set II, write M/IM/I and N/IN/I for the matroid contractions, and let r(M/I)r(M/I) and r(N/I)r(N/I) denote their ranks. An (M,N)(M,N)-hindrance is an (M,N)(M,N)-wave that does not span N.spanM(I)N.\mathsf{span}_M(I), where an (M,N)(M,N)-wave is a set in I(M)I(N.spanM(I))\mathcal{I}(M)\cap\mathcal{I}(N.\mathsf{span}_M(I)). Wasteful partial linkage hindrance conjecture. If there is an II(M)I(N)I\in\mathcal{I}(M)\cap\mathcal{I}(N) such that

r(M/I)<r(N/I),r(M/I)<r(N/I),

then (M,N)(M,N) is hindered. This proposes a criterion for detecting hindrance in the problem of finding an MM-independent spanning set of NN; the supplied text does not indicate whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Attila Joó, “Hindrance from a wasteful partial linkage”, arXiv:2410.19583 (2025).

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