Add known results and bibliography

Introduce a "Known results" section discussing β(M1∩M2), state the
Aharoni–Berger conjectures and the Davies–McDiarmid theorem, and add
three bibliography entries (emlektabla16_2024, rainbow bases SIAM,
berczi_partitioning_2024).

notes/main.md

@@ -2,6 +2,8 @@
 title: Matroid Rainbow Base Cover
 ----
 
+This problem is proposed in [@emlektabla16_2024].
+
 Let $M=(E,\mathcal B)$  be a rank-$r$ matroid whose ground set decomposes into two disjoint bases. Furthermore,
 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.
@@ -23,4 +25,27 @@ Then $M_1$ and $M_2$ has four common bases that cover each element exactly twice
 :::
 
 Let $M_1$ be the partition matroid of colors and let $M_2$ be $M$.
-If [@conj-doublecover] is true, then 3 common bases will be enough to cover $E$.
+If [@conj-doublecover] is true, then 3 common bases will be enough to cover $E$.
+
+# Known results
+
+Let $\beta(M)$ be the covering number of matroid $M$. Given two matroids $M_1,M_2$ on the same ground set, $\beta(M_1\cap M_2)$ is the minimum number of common independent sets needed to cover the ground set.
+[@conj-doublecover] is basically asking for $\beta(M_1\cap M_2)$ when the ground set $E$ is partitioned into 2 common bases of $M_1$ and $M_2$.
+
+## Results and conjectures on $\beta(M_1\cap M_2)$
+
+One can see that $\beta(M_1\cap M_2)\geq \min \set{\beta(M_1),\beta(M_2)}$. Aharoni and Berger showed that $\beta(M_1\cap M_2)\leq 2 \max \set{\beta(M_1),\beta(M_2)}$
+
+::: {.Conjecture title="Aharoni and Berger"}
+Let $M_1$ and $M_2$ be two matroids on the same ground set.
+
+1. If $\beta(M_1)\neq \beta(M_2)$, then $\beta(M_1\cap M_2)=\max \set{\beta(M_1),\beta(M_2)}$.
+2. If $\beta(M_1)= \beta(M_2)$, then $\beta(M_1\cap M_2)\leq \max \set{\beta(M_1),\beta(M_2)}+1$.
+:::
+
+::: {.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.

reference.bib

@@ -0,0 +1,30 @@
+@misc{emlektabla16_2024,
+  title        = {16th Eml{\'e}kt{\'a}bla Workshop},
+  year         = {2024},
+  howpublished = {\url{https://users.renyi.hu/~emlektab/emlektabla16.pdf}},
+  note         = {Workshop PDF; accessed 2026-03-26}
+}
+
+@article{noauthor_rainbow_nodate,
+	title = {Rainbow {Bases} in {Matroids}},
+	url = {https://epubs.siam.org/doi/10.1137/22M1516750},
+	abstract = {Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function such that every matroid that can be factorized into bases for some can be covered by rainbow bases if every partition class contains at most 2 elements.},
+	language = {en},
+	urldate = {2026-03-26},
+	journal = {SIAM Journal on Discrete Mathematics},
+}
+
+@article{berczi_partitioning_2024,
+	title = {Partitioning into common independent sets via relaxing strongly base orderability},
+	volume = {202},
+	issn = {00973165},
+	url = {https://linkinghub.elsevier.com/retrieve/pii/S0097316523000857},
+	doi = {10.1016/j.jcta.2023.105817},
+	language = {en},
+	urldate = {2026-03-26},
+	journal = {Journal of Combinatorial Theory, Series A},
+	author = {Bérczi, Kristóf and Schwarcz, Tamás},
+	month = feb,
+	year = {2024},
+	pages = {105817},
+}