From b7e87b859e2472e462cf62ecb556b8654ecdf3dc Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Wed, 11 Mar 2026 09:55:28 +0800
Subject: [PATCH] Update main.tex

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 main.tex | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)

diff --git a/main.tex b/main.tex
index 775c245..c3769b2 100644
--- a/main.tex
+++ b/main.tex
@@ -294,6 +294,25 @@ Given a hypergraph with $<k$ components, we can add at most $O(|V|)$ dummy hyper
 
 \autoref{thm:kcut} reduces minimum $k$-cut problem on hypergraphs to minimum $k$-cocycle on hypergraphic matroids. Now we try to apply the ideal base packing framework on hypergraphic matroids.
 
+\begin{lemma}
+	Let $H$ be a parititon-connected hypergraph and let $M$ be the hypergraphic matroid on $H$.
+	There is a distribution $\mathcal D$ on $k$-cocycles such that for any base in the ideal base packing, the expected size of their intersection is at most $2k$.
+\end{lemma}
+\begin{proof}
+	Let flat $F\in \argmin_{F'\subset E}\frac{|E-F'|}{r(E)-r(F')}$ be the optimal set for matroid strength.
+	Each flat in $M$ induces a vertex partition $\mathcal P_F$ by its forest representation.
+	One can see that $|\mathcal P|=r(E)-r(F)+1$. Now we randomly pick $r(E)-r(F)+1-k$ parts and merge them into a new component.
+	Let $\mathcal P'$ be the resulting partition and let $B$ be any ideal base of $M$.
+	We have $e_B(\mathcal P)=r(E)-r(F)$ since $B$ is ideal. Now we count the expected number of $e_B(\mathcal P')$,
+	\[
+    	\begin{aligned}
+    		\E_{\mathcal D}[e_B(\mathcal P')] &\leq e_B(\mathcal P)\Pr[\text{a hyperedge is not contained in the new part}] \\
+					&\leq (r(E)-r(F))\cdot ???
+        \end{aligned}
+	\]
+	Note that the first inequality is because $\mathcal P'$ may induce a $\geq k$-cocycle, the second one ...
+\end{proof}
+
 \section{Greedy base packing}
 
 \section{Principal sequence + KT contraction}