From 0e4b2e9d268849fd9518471e85825fbd4e9711e8 Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Thu, 4 Dec 2025 13:47:34 +0800
Subject: [PATCH] z

---
 main.tex | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/main.tex b/main.tex
index 8febbe0..0f5834a 100644
--- a/main.tex
+++ b/main.tex
@@ -178,8 +178,10 @@ for any base $B$ of $M'$, the expected size of intersection is at most $O(k')$.
 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.
 
-\paragraph{Nice properties}
-$\cl(F+e)\setminus F$ is a partition of $E\setminus F$ for general matroids.
+\begin{proposition}
+Let $M=(E,\mathcal I)$ be a matroids and let $F$ be a flat of $M$.
+Then $\cl(F+e)\setminus F$ is a partition of $E\setminus F$.
+\end{proposition}
 
 \subsection{Support size}
 
@@ -192,7 +194,7 @@ The strategy is to first find a $\geq k$-cocycle using the utilization algorithm
 Note that LP gives an ideal tree packing with $O(m)$ support size.
 
 \subsection{Rigidity matroids}
-\begin{conjecture}\label{conj:idealrigidbase}
+\begin{conjecture}
 Let $M$ be a connected 2D rigidity matroid on graph $G=(V,E)$. Let $F^*$ be the optimal flat for strength $F^*=\argmin_{F\subset E}\frac{c(E\setminus F)}{r(E)-r(F)}$.
 Let $X\subset E\setminus F^*$ be a independent set with rank $r(E)-r(F^*)$. Then for any maximal independent set $B_{F^*}\subset F^*$, $X\cup B_{F^*}$ is a base of $M$.
 \end{conjecture}