Large k-Connected Set Persists In A Weakly 4-Connected Minor

Claim/Theorem

Let \(Z\) be an \(n\)-element \(k\)-connected set in a matroid \(M\), with

\[ n\ge 3k-5 \qquad\text{and}\qquad k\ge 4. \]

Then \(M\) has a weakly \(4\)-connected minor \(N\) that contains an \(n\)-element \(k\)-connected set.

Thus a large highly connected set can be transferred into a weakly \(4\)-connected minor without shrinking the set or lowering its \(k\)-connectivity.

In particular, if one could prove that the relevant Quantum Tanner parity-check matroid contains a large \(k\)-connected set with \(k\) growing with \(n\), then one would obtain a weakly \(4\)-connected minor carrying the same high-order connectivity.

Dependencies

• [[tangle-breadth-gives-k-connected-set.md]]

Conflicts/Gaps

• The theorem preserves an already-existing \(k\)-connected set; it does not create one from branchwidth alone.
• Weakly \(4\)-connectedness remains only a background structural condition. The real quantitative force is in the preserved \(k\)-connected set.
• This still works at the level of minors, not necessarily the original parity-check matroid of the code.

Sources

• 10.37236/12467