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Add rainbow-base counterexample search and notes
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notes/counterexample-search-2026-03-26.md
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notes/counterexample-search-2026-03-26.md
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title: Counterexample Search Notes
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---
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# Goal
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Try to find a counterexample to the rainbow-base-cover conjecture:
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> 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.
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I also tracked the stronger special case of the double-cover conjecture in the partition-matroid setting:
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> For the rainbow bases, are there always four of them covering each element exactly twice?
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# Search Design
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## Encoding
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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.
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For a matroid `M`, I tested:
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1. whether `M` has two disjoint bases;
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2. for each pairing, which of the `2^r` transversals are actual bases;
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3. whether some 3 rainbow bases cover all `2r` elements;
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4. whether some 4 rainbow bases cover every element exactly twice.
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This was implemented in [search_rainbow_counterexample.py](/Users/congyu/rainbow_base_cover/search_rainbow_counterexample.py).
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## Exhaustive part
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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:
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- `r = 1`: `1` pairing
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- `r = 2`: `3` pairings
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- `r = 3`: `15` pairings
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- `r = 4`: `105` pairings
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## Additional rank-5 probes
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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:
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- random rank-5 linear matroids over `GF(2), GF(3), GF(4), GF(5)`;
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- all simple graphic matroids coming from 8-edge graphs on 5 vertices.
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# Results
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## Exhaustive search for `r <= 4`
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No counterexample appeared.
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| rank `r` | matroids checked | qualifying matroids | max observed minimum cover | any 3-cover failure? | any 4-double-cover failure? |
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|---|---:|---:|---:|---|---|
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| 1 | 2 | 1 | 2 | no | no |
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| 2 | 7 | 3 | 2 | no | no |
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| 3 | 38 | 17 | 3 | no | no |
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| 4 | 940 | 730 | 3 | no | no |
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So:
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- the rainbow-cover conjecture holds for every matroid in Sage's complete database with `2r <= 8`;
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- in the same range, the stronger partition-matroid double-cover statement also holds.
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## Explicit rank-4 witness requiring 3 bases
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The first rank-4 example I found with minimum cover number exactly `3` is
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- matroid: `all_n08_r04_#493`
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- pairing: `((0,5),(1,4),(2,3),(6,7))`
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For this pairing there are exactly `8` rainbow bases:
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`(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)`.
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This instance still has:
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- minimum rainbow cover size `= 3`;
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- a 4-rainbow exact double cover.
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So the search really is reaching the sharp bound `3`, not just easy cases with cover number `2`.
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## Graphic search
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The known lower-bound example `K_4` is reproduced computationally:
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- all simple graphs on 4 vertices with 6 edges: `1` graph checked;
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- maximum minimum cover number: `3`;
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- no 3-cover failure;
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- no 4-double-cover failure.
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I also checked all simple 8-edge graphs on 5 vertices:
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- graphs checked: `45`;
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- connected graphs: `45`;
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- qualifying graphic matroids: `45`;
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- maximum minimum cover number: `3`;
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- no 3-cover failure;
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- no 4-double-cover failure.
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One concrete simple graphic rank-4 witness with minimum cover `3` is the graph with edge set
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`((0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3))`
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and pairing
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`((0,3),(1,5),(2,4),(6,7))`.
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## Random rank-5 linear search
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No sampled rank-5 linear matroid produced a counterexample.
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Finished runs:
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| field | samples | distinct matroids checked | qualifying matroids | max observed minimum cover | any 3-cover failure? | any 4-double-cover failure? |
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|---|---:|---:|---:|---:|---|---|
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| `GF(2)` | 200 | 200 | 92 | 3 | no | no |
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| `GF(2)` | 500 | 500 | 245 | 3 | no | no |
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| `GF(3)` | 200 | 200 | 180 | 3 | no | no |
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| `GF(4)` | 200 | 200 | 99 | 3 | no | no |
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| `GF(5)` | 200 | 200 | 200 | 2 | no | no |
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# Observations
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## Small-rank evidence is strong
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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.
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## Rank 4 already has nontrivial tight examples
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The bound `3` is still best possible in rank 4: there are pairings where 2 rainbow bases do not suffice, but 3 do.
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## The partition-matroid special case looks robust
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At least computationally, the partition-matroid case of the double-cover conjecture behaves better than expected:
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- exhaustive success for all matroids on 8 elements of rank 4;
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- no failures in the rank-5 linear samples over four small fields;
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- no failures in the checked graphic families.
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# What I would try next
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## More targeted rank-5 search
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The next most plausible places to look are:
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1. exhaustive or semi-exhaustive graphic search on 10 edges and 6 vertices;
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2. sparse paving matroids of rank 5 on 10 elements;
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3. biased random linear constructions, especially sparse or highly structured matrices rather than dense uniform random ones;
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4. direct search for the stronger four-basis exact double-cover failure, since that might break before the three-cover statement does.
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## Structural reformulation
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For a fixed pairing, the rainbow bases form a subset `F ⊆ {0,1}^r`. Then:
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- 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;
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- 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.
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This reformulation may be a better starting point for a structural attack than thinking directly in matroid language.
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# Current Status
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I did not find a counterexample.
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The strongest completed evidence from this turn is:
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- exhaustive verification for all rank-`r` matroids on `2r` elements with `r <= 4`;
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- exhaustive verification for simple graphic rank-4 instances on 8 edges;
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- no sampled failure among several hundred rank-5 linear matroids over `GF(2), GF(3), GF(4), GF(5)`.
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