Tangle Breadth Gives A Large k-Connected Set

Claim/Theorem

Let \(T\) be a tangle of order \(k\) in a matroid \(M\), and let the breadth of \(T\) be \(t\), meaning that the tangle matroid \(M_T\) contains a largest spanning uniform submatroid of size \(t\).

Then:

1. \(M\) contains a \(t\)-element \(k\)-connected set \(Z\) such that

\[ M_T|Z \cong U_{k-1,t}. \]

1. If additionally

\[ t\ge 3k-5 \]

and \(k\ge 3\), then the tangle \(T\) is generated by the set \(Z\).

Here \(Z\) being \(k\)-connected means that for all \(A\subseteq E(M)\),

\[ \lambda_M(A)\ge \min\{|A\cap Z|,\ |Z-A|,\ k-1\}. \]

So tangle breadth, not just tangle order, is the intrinsic quantity that yields a genuinely large highly connected set.

Dependencies

• None.

Conflicts/Gaps

• This theorem does not provide a lower bound on breadth from order or branchwidth alone.
• Therefore it does not upgrade the existing Omega(n/log n) branchwidth theorem by itself.
• For Conjecture 3, it identifies a sharper missing ingredient: prove that the relevant Quantum Tanner parity-check matroids have tangles of large breadth, not merely large order.

Sources

• 10.37236/12467