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main.tex

@@ -160,9 +160,24 @@ One can carefully design the distributions for contractions so instead of contra
 We uniformly and randomly choose $k-k'+1$ parts in $\mathcal P_F$ and merge them into a big part.
 Denote the resulting partition by $\mathcal P_{F'}$.
 Let $T$ be a spanning tree in the support of ideal tree packing. 
-Note that the number of inter-component edges of $T$ in $\mathcal P_F$ is $k$.
+Recall that the number of inter-component edges of $T$ in $\mathcal P_F$ is $k$.
+Then we have
+\[
+\E_{F'}[|T\setminus F'|]=k\brack{1-\paren{\frac{k+1-k'}{k+1}}^2}\leq 2k'.
+\]
 \end{proof}
 
+For general matroids, we want to show the following.
+
+\begin{conjecture}
+Let $M'$ be the contraction $M/F^*$. The rank of $M'$ is $k$. 
+Given a positive integer $k'<k$, then there exists a distribution on $k'$-cocycles such that
+for any base $B$ of $M'$, the expected size of intersection is at most $O(k')$.
+\end{conjecture}
+
+A counterexample would be uniform matroids $U_{2n,n}$.
+The size of every $k'$-cocycle is $k'+n$, and for any $k'$-cocycle there are bases using $O(n)$ elements in the cocycle.
+
 \subsection{Support size}
 
 Recall that we construct the ideal base packing recursively. Suppose that the ideal base packing for $M|F^*$ contains $n$ bases and let $m$ be the size of support of the optimal base packing of $M$. Then the number of bases in the ideal base packing of $M$ is $nm$. Note that $m$ is upperbounded by the $|E|$ since the number of constraints is at most $|E|$ in the tree packing LP.