z

notes/2026-03-26-counterexample-search.md

@@ -1,5 +1,5 @@
 ---
-title: Counterexample Search Notes
+title: codex - Counterexample Search Notes
 ---
 
 # Goal

notes/2026-03-26-proof-attempt.md

@@ -1,5 +1,5 @@
 ---
-title: Proof Attempt and the Strongly Base Orderable Case
+title: codex - Proof Attempt and the Strongly Base Orderable Case
 ---
 
 # Status

notes/main.md

@@ -8,7 +8,7 @@ Let $M=(E,\mathcal B)$  be a rank-$r$ matroid whose ground set decomposes into t
 assume that $E$ is colored by $r$ colors, each color appearing exactly twice. A basis of $M$ is called rainbow if
 it does not contain two elements of the same color.
 
-::: Problem
+::: {.Problem #prob-rainbowbasecover}
 What is the minimum number of rainbow bases needed to cover $E$?
 :::
 
@@ -21,7 +21,7 @@ Currently known bounds:
 
 ::: {.Conjecture #conj-doublecover}
 Let $M_1$ and $M_2$ be two matroid on the same ground set $E$. Assume that $E$ decomposes into two bases in both of them.
-Then $M_1$ and $M_2$ has four common bases that cover each element exactly twice.
+Then $M_1$ and $M_2$ has four common bases (allow repetition) that cover each element exactly twice.
 :::
 
 Let $M_1$ be the partition matroid of colors and let $M_2$ be $M$.
@@ -43,9 +43,35 @@ Let $M_1$ and $M_2$ be two matroids on the same ground set.
 2. If $\beta(M_1)= \beta(M_2)$, then $\beta(M_1\cap M_2)\leq \max \set{\beta(M_1),\beta(M_2)}+1$.
 :::
 
+Cases that have been verified:
+
+- If $\beta(M_1)=\beta(M_2)=2$ then $\beta(M_1\cap M_2)=3$.^[Note that this case does not give an answer to [@prob-rainbowbasecover] since the covering in $\beta(M_1\cap M_2)$ uses common independent sets instead of common bases.]
+- If $\beta(M_1)=2,\beta(M_2)=3$, then $\beta(M_1\cap M_2)\leq 4$
+- ...
+
 ::: {.Theorem title="Davies and McDiarmid"}
 Let $M_1$ and $M_2$ be strongly base orderable matroids on the same ground set.
 Then $\beta(M_1\cap M_2)= \max \set{\beta(M_1),\beta(M_2)}$.
 :::
 
-See [@berczi_partitioning_2024] for refs.
+See [@berczi_partitioning_2024] for refs. Bérczi and Schwarcz generalized SBO using the following definition.
+
+::: {.Definition title="strongly base reorderable"}
+A matroid is called **strongly base reorderable**(SBRO) if for any pair $A,B$ of bases, there exists bases $A',B'$ and a bijection $\phi:A'\setminus B'\to B'\setminus A'$ such that $A'\cap B'=A\cap B,A'\cup B'=A\cup B$, and $(A'\setminus X)\cup \phi(X)$ is a basis for every $X\subset A'\setminus B'$.
+:::
+
+::: Lemma
+The class of SRBO matroids is complete(closed under taking minors, duals, direct sums, truncations and induction by directed graphs).
+:::
+
+They also showed that SBRO is sufficient for $\beta(M_1\cap M_2)=\max \set{\beta(M_1),\beta(M_2)}$.
+
+::: Theorem
+Let $M_1$ and $M_2$ be strongly base reorderable matroids on the same ground set.
+Then $\beta(M_1\cap M_2)= \max \set{\beta(M_1),\beta(M_2)}$.
+:::
+
+## Circuit cover
+
+Let $M=(E,\mathcal B)$ be a matroid and let $A,B\in \mathcal B$ be two bases of $M$.
+Define a graph $G=(A\Delta B, F)$.