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

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title: Counterexample Search Notes
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# Goal

Try to find a counterexample to the rainbow-base-cover conjecture:

> If a rank-`r` matroid on `2r` elements decomposes into two disjoint bases, and the elements are colored by `r` colors each used twice, then three rainbow bases cover the ground set.

I also tracked the stronger special case of the double-cover conjecture in the partition-matroid setting:

> For the rainbow bases, are there always four of them covering each element exactly twice?

# Search Design

## Encoding

Fix a pairing of the `2r` elements into color classes. A rainbow basis chooses exactly one element from each pair, so for fixed coloring there are at most `2^r` rainbow candidates.

For a matroid `M`, I tested:

1. whether `M` has two disjoint bases;
2. for each pairing, which of the `2^r` transversals are actual bases;
3. whether some 3 rainbow bases cover all `2r` elements;
4. whether some 4 rainbow bases cover every element exactly twice.

This was implemented in [search_rainbow_counterexample.py](/Users/congyu/rainbow_base_cover/search_rainbow_counterexample.py).

## Exhaustive part

Sage's `AllMatroids(2r, r)` is available up to `r = 4`, so I checked all unlabeled rank-`r` matroids on `2r` elements for `r = 1,2,3,4`, and all pairings of the ground set:

- `r = 1`: `1` pairing
- `r = 2`: `3` pairings
- `r = 3`: `15` pairings
- `r = 4`: `105` pairings

## Additional rank-5 probes

Since Sage does not exhaust all rank-5 matroids on 10 elements, I also sampled random linear matroids over small fields, and exhaustively checked some graphic families:

- random rank-5 linear matroids over `GF(2), GF(3), GF(4), GF(5)`;
- all simple graphic matroids coming from 8-edge graphs on 5 vertices.

# Results

## Exhaustive search for `r <= 4`

No counterexample appeared.

| rank `r` | matroids checked | qualifying matroids | max observed minimum cover | any 3-cover failure? | any 4-double-cover failure? |
|---|---:|---:|---:|---|---|
| 1 | 2 | 1 | 2 | no | no |
| 2 | 7 | 3 | 2 | no | no |
| 3 | 38 | 17 | 3 | no | no |
| 4 | 940 | 730 | 3 | no | no |

So:

- the rainbow-cover conjecture holds for every matroid in Sage's complete database with `2r <= 8`;
- in the same range, the stronger partition-matroid double-cover statement also holds.

## Explicit rank-4 witness requiring 3 bases

The first rank-4 example I found with minimum cover number exactly `3` is

- matroid: `all_n08_r04_#493`
- pairing: `((0,5),(1,4),(2,3),(6,7))`

For this pairing there are exactly `8` rainbow bases:

`(0,1,3,6)`, `(0,1,3,7)`, `(0,2,4,6)`, `(0,2,4,7)`, `(1,2,5,6)`, `(1,2,5,7)`, `(3,4,5,6)`, `(3,4,5,7)`.

This instance still has:

- minimum rainbow cover size `= 3`;
- a 4-rainbow exact double cover.

So the search really is reaching the sharp bound `3`, not just easy cases with cover number `2`.

## Graphic search

The known lower-bound example `K_4` is reproduced computationally:

- all simple graphs on 4 vertices with 6 edges: `1` graph checked;
- maximum minimum cover number: `3`;
- no 3-cover failure;
- no 4-double-cover failure.

I also checked all simple 8-edge graphs on 5 vertices:

- graphs checked: `45`;
- connected graphs: `45`;
- qualifying graphic matroids: `45`;
- maximum minimum cover number: `3`;
- no 3-cover failure;
- no 4-double-cover failure.

One concrete simple graphic rank-4 witness with minimum cover `3` is the graph with edge set

`((0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3))`

and pairing

`((0,3),(1,5),(2,4),(6,7))`.

## Random rank-5 linear search

No sampled rank-5 linear matroid produced a counterexample.

Finished runs:

| field | samples | distinct matroids checked | qualifying matroids | max observed minimum cover | any 3-cover failure? | any 4-double-cover failure? |
|---|---:|---:|---:|---:|---|---|
| `GF(2)` | 200 | 200 | 92 | 3 | no | no |
| `GF(2)` | 500 | 500 | 245 | 3 | no | no |
| `GF(3)` | 200 | 200 | 180 | 3 | no | no |
| `GF(4)` | 200 | 200 | 99 | 3 | no | no |
| `GF(5)` | 200 | 200 | 200 | 2 | no | no |

# Observations

## Small-rank evidence is strong

Up through rank 4, the search is exhaustive, not heuristic. In that range I found no obstruction even to the stronger four-basis exact double-cover property.

## Rank 4 already has nontrivial tight examples

The bound `3` is still best possible in rank 4: there are pairings where 2 rainbow bases do not suffice, but 3 do.

## The partition-matroid special case looks robust

At least computationally, the partition-matroid case of the double-cover conjecture behaves better than expected:

- exhaustive success for all matroids on 8 elements of rank 4;
- no failures in the rank-5 linear samples over four small fields;
- no failures in the checked graphic families.

# What I would try next

## More targeted rank-5 search

The next most plausible places to look are:

1. exhaustive or semi-exhaustive graphic search on 10 edges and 6 vertices;
2. sparse paving matroids of rank 5 on 10 elements;
3. biased random linear constructions, especially sparse or highly structured matrices rather than dense uniform random ones;
4. direct search for the stronger four-basis exact double-cover failure, since that might break before the three-cover statement does.

## Structural reformulation

For a fixed pairing, the rainbow bases form a subset `F ⊆ {0,1}^r`. Then:

- 3-cover means there exist `x,y,z in F` such that in every coordinate, not all of `x_i,y_i,z_i` are equal;
- exact 4-double-cover means there exist `x_1,x_2,x_3,x_4 in F` such that every coordinate has exactly two `0`s and two `1`s.

This reformulation may be a better starting point for a structural attack than thinking directly in matroid language.

# Current Status

I did not find a counterexample.

The strongest completed evidence from this turn is:

- exhaustive verification for all rank-`r` matroids on `2r` elements with `r <= 4`;
- exhaustive verification for simple graphic rank-4 instances on 8 edges;
- no sampled failure among several hundred rank-5 linear matroids over `GF(2), GF(3), GF(4), GF(5)`.