Wiki
Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
direct.json
json
{
"coverage": "exact constraint inclusion-exclusion and four-block recurrence",
"total_colorings_represented": 17421,
"records": [
{
"k": 2,
"claimed_minimum": 3,
"rows": [
{
"N": 0,
"ap_count": 0,
"colorings": 1,
"avoiding_colorings": 1,
"first_avoiding_coloring": []
},
{
"N": 1,
"ap_count": 0,
"colorings": 2,
"avoiding_colorings": 2,
"first_avoiding_coloring": [
-1
]
},
{
"N": 2,
"ap_count": 1,
"colorings": 4,
"avoiding_colorings": 2,
"first_avoiding_coloring": [
-1,
1
]
},
{
"N": 3,
"ap_count": 3,
"colorings": 8,
"avoiding_colorings": 0,
"first_avoiding_coloring": null
}
]
},
{
"k": 3,
"claimed_minimum": 9,
"rows": [
{
"N": 0,
"ap_count": 0,
"colorings": 1,
"avoiding_colorings": 1,
"first_avoiding_coloring": []
},
{
"N": 1,
"ap_count": 0,
"colorings": 2,
"avoiding_colorings": 2,
"first_avoiding_coloring": [
-1
]
},
{
"N": 2,
"ap_count": 0,
"colorings": 4,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1
]
},
{
"N": 3,
"ap_count": 1,
"colorings": 8,
"avoiding_colorings": 6,
"first_avoiding_coloring": [
-1,
-1,
1
]
},
{
"N": 4,
"ap_count": 2,
"colorings": 16,
"avoiding_colorings": 10,
"first_avoiding_coloring": [
-1,
-1,
1,
-1
]
},
{
"N": 5,
"ap_count": 4,
"colorings": 32,
"avoiding_colorings": 14,
"first_avoiding_coloring": [
-1,
-1,
1,
-1,
-1
]
},
{
"N": 6,
"ap_count": 6,
"colorings": 64,
"avoiding_colorings": 20,
"first_avoiding_coloring": [
-1,
-1,
1,
-1,
-1,
1
]
},
{
"N": 7,
"ap_count": 9,
"colorings": 128,
"avoiding_colorings": 16,
"first_avoiding_coloring": [
-1,
-1,
1,
-1,
-1,
1,
1
]
},
{
"N": 8,
"ap_count": 12,
"colorings": 256,
"avoiding_colorings": 6,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1,
1
]
},
{
"N": 9,
"ap_count": 16,
"colorings": 512,
"avoiding_colorings": 0,
"first_avoiding_coloring": null
}
]
},
{
"k": 4,
"claimed_minimum": 13,
"rows": [
{
"N": 0,
"ap_count": 0,
"colorings": 1,
"avoiding_colorings": 1,
"first_avoiding_coloring": []
},
{
"N": 1,
"ap_count": 0,
"colorings": 2,
"avoiding_colorings": 2,
"first_avoiding_coloring": [
-1
]
},
{
"N": 2,
"ap_count": 0,
"colorings": 4,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1
]
},
{
"N": 3,
"ap_count": 0,
"colorings": 8,
"avoiding_colorings": 8,
"first_avoiding_coloring": [
-1,
-1,
-1
]
},
{
"N": 4,
"ap_count": 1,
"colorings": 16,
"avoiding_colorings": 6,
"first_avoiding_coloring": [
-1,
-1,
1,
1
]
},
{
"N": 5,
"ap_count": 2,
"colorings": 32,
"avoiding_colorings": 6,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1
]
},
{
"N": 6,
"ap_count": 3,
"colorings": 64,
"avoiding_colorings": 6,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1
]
},
{
"N": 7,
"ap_count": 5,
"colorings": 128,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1
]
},
{
"N": 8,
"ap_count": 7,
"colorings": 256,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1,
1
]
},
{
"N": 9,
"ap_count": 9,
"colorings": 512,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1,
1,
-1
]
},
{
"N": 10,
"ap_count": 12,
"colorings": 1024,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1,
1,
-1,
-1
]
},
{
"N": 11,
"ap_count": 15,
"colorings": 2048,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1,
1,
-1,
-1,
1
]
},
{
"N": 12,
"ap_count": 18,
"colorings": 4096,
"avoiding_colorings": 4,
"first_avoiding_coloring": [
-1,
-1,
1,
1,
-1,
-1,
1,
1,
-1,
-1,
1,
1
]
},
{
"N": 13,
"ap_count": 22,
"colorings": 8192,
"avoiding_colorings": 0,
"first_avoiding_coloring": null
}
]
}
],
"certificates": [
{
"k": 2,
"rows": [
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 1,
"coefficient_polynomial": [
[
0,
1
]
],
"first_witness_prefix_counts": [],
"N": 0,
"endpoint_progressions": []
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 1,
"coefficient_polynomial": [
[
1,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 1,
"plus_count": 1
}
],
"N": 1,
"endpoint_progressions": []
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 2,
"coefficient_polynomial": [
[
1,
-1
],
[
2,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 1,
"plus_count": 1
},
{
"vertex": 1,
"minus_count": 0,
"plus_count": 1
}
],
"N": 2,
"endpoint_progressions": [
[
0,
1
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 5,
"coefficient_polynomial": [
[
1,
2
],
[
2,
-3
],
[
3,
1
]
],
"first_witness_prefix_counts": [],
"N": 3,
"endpoint_progressions": [
[
0,
1
],
[
0,
2
],
[
1,
2
]
]
}
]
},
{
"k": 3,
"rows": [
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 1,
"coefficient_polynomial": [
[
0,
1
]
],
"first_witness_prefix_counts": [],
"N": 0,
"endpoint_progressions": []
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 1,
"coefficient_polynomial": [
[
1,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 1,
"plus_count": 1
}
],
"N": 1,
"endpoint_progressions": []
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 1,
"coefficient_polynomial": [
[
2,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 2,
"plus_count": 2
},
{
"vertex": 1,
"minus_count": 1,
"plus_count": 1
}
],
"N": 2,
"endpoint_progressions": []
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 2,
"coefficient_polynomial": [
[
1,
-1
],
[
3,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 3,
"plus_count": 3
},
{
"vertex": 1,
"minus_count": 1,
"plus_count": 2
},
{
"vertex": 2,
"minus_count": 0,
"plus_count": 1
}
],
"N": 3,
"endpoint_progressions": [
[
0,
1,
2
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 4,
"coefficient_polynomial": [
[
1,
1
],
[
2,
-2
],
[
4,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 5,
"plus_count": 5
},
{
"vertex": 1,
"minus_count": 2,
"plus_count": 3
},
{
"vertex": 2,
"minus_count": 0,
"plus_count": 2
},
{
"vertex": 3,
"minus_count": 1,
"plus_count": 1
}
],
"N": 4,
"endpoint_progressions": [
[
0,
1,
2
],
[
1,
2,
3
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 10,
"coefficient_polynomial": [
[
1,
-1
],
[
2,
4
],
[
3,
-4
],
[
5,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 7,
"plus_count": 7
},
{
"vertex": 1,
"minus_count": 3,
"plus_count": 4
},
{
"vertex": 2,
"minus_count": 0,
"plus_count": 3
},
{
"vertex": 3,
"minus_count": 2,
"plus_count": 1
},
{
"vertex": 4,
"minus_count": 1,
"plus_count": 1
}
],
"N": 5,
"endpoint_progressions": [
[
0,
1,
2
],
[
0,
2,
4
],
[
1,
2,
3
],
[
2,
3,
4
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 19,
"coefficient_polynomial": [
[
1,
-2
],
[
3,
7
],
[
4,
-6
],
[
6,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 10,
"plus_count": 10
},
{
"vertex": 1,
"minus_count": 4,
"plus_count": 6
},
{
"vertex": 2,
"minus_count": 0,
"plus_count": 4
},
{
"vertex": 3,
"minus_count": 2,
"plus_count": 2
},
{
"vertex": 4,
"minus_count": 1,
"plus_count": 1
},
{
"vertex": 5,
"minus_count": 0,
"plus_count": 1
}
],
"N": 6,
"endpoint_progressions": [
[
0,
1,
2
],
[
0,
2,
4
],
[
1,
2,
3
],
[
1,
3,
5
],
[
2,
3,
4
],
[
3,
4,
5
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 43,
"coefficient_polynomial": [
[
1,
4
],
[
2,
-12
],
[
3,
5
],
[
4,
11
],
[
5,
-9
],
[
7,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 8,
"plus_count": 8
},
{
"vertex": 1,
"minus_count": 4,
"plus_count": 4
},
{
"vertex": 2,
"minus_count": 0,
"plus_count": 4
},
{
"vertex": 3,
"minus_count": 1,
"plus_count": 3
},
{
"vertex": 4,
"minus_count": 1,
"plus_count": 0
},
{
"vertex": 5,
"minus_count": 0,
"plus_count": 1
},
{
"vertex": 6,
"minus_count": 0,
"plus_count": 1
}
],
"N": 7,
"endpoint_progressions": [
[
0,
1,
2
],
[
0,
2,
4
],
[
0,
3,
6
],
[
1,
2,
3
],
[
1,
3,
5
],
[
2,
3,
4
],
[
2,
4,
6
],
[
3,
4,
5
],
[
4,
5,
6
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 101,
"coefficient_polynomial": [
[
1,
-3
],
[
2,
21
],
[
3,
-39
],
[
4,
17
],
[
5,
15
],
[
6,
-12
],
[
8,
1
]
],
"first_witness_prefix_counts": [
{
"vertex": 0,
"minus_count": 3,
"plus_count": 3
},
{
"vertex": 1,
"minus_count": 1,
"plus_count": 2
},
{
"vertex": 2,
"minus_count": 0,
"plus_count": 1
},
{
"vertex": 3,
"minus_count": 0,
"plus_count": 1
},
{
"vertex": 4,
"minus_count": 1,
"plus_count": 0
},
{
"vertex": 5,
"minus_count": 1,
"plus_count": 0
},
{
"vertex": 6,
"minus_count": 0,
"plus_count": 1
},
{
"vertex": 7,
"minus_count": 0,
"plus_count": 1
}
],
"N": 8,
"endpoint_progressions": [
[
0,
1,
2
],
[
0,
2,
4
],
[
0,
3,
6
],
[
1,
2,
3
],
[
1,
3,
5
],
[
1,
4,
7
],
[
2,
3,
4
],
[
2,
4,
6
],
[
3,
4,
5
],
[
3,
5,
7
],
[
4,
5,
6
],
[
5,
6,
7
]
]
},
{
"route": "inclusion-exclusion over monochromatic constraints",
"nonzero_partition_terms": 264,
"coefficient_polynomial": [
[
1,
-10
],
[
2,
5
],
[
3,
58
],
[
4,
-101
],
[
5,
42
],
[
6,
21
],
[
7,
-16
],
[
9,
1
]
],
"first_witness_prefix_counts": [],
"N": 9,
"endpoint_progressions": [
[
0,
1,
2
],
[
0,
2,
4
],
[
0,
3,
6
],
[
0,
4,
8
],
[
1,
2,
3
],
[
1,
3,
5
],
[
1,
4,
7
],
[
2,
3,
4
],
[
2,
4,
6
],
[
2,
5,
8
],
[
3,
4,
5
],
[
3,
5,
7
],
[
4,
5,
6
],
[
4,
6,
8
],
[
5,
6,
7
],
[
6,
7,
8
]
]
}
]
},
{
"k": 4,
"rows": [
{
"route": "no four-point progression",
"seeds": [],
"N": 0,
"endpoint_progressions": []
},
{
"route": "no four-point progression",
"seeds": [],
"N": 1,
"endpoint_progressions": []
},
{
"route": "no four-point progression",
"seeds": [],
"N": 2,
"endpoint_progressions": []
},
{
"route": "no four-point progression",
"seeds": [],
"N": 3,
"endpoint_progressions": []
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
-1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
-1,
1,
-1
],
[
1,
1,
-1,
-1
]
],
"N": 4,
"endpoint_progressions": [
[
0,
1,
2,
3
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
-1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
-1,
1,
-1
],
[
1,
1,
-1,
-1
]
],
"N": 5,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
1,
2,
3,
4
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
-1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
-1,
1,
-1
],
[
1,
1,
-1,
-1
]
],
"N": 6,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
1,
2,
3,
4
],
[
2,
3,
4,
5
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
1,
-1,
-1
]
],
"N": 7,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
1,
2,
3,
4
],
[
2,
3,
4,
5
],
[
3,
4,
5,
6
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
1,
-1,
-1
]
],
"N": 8,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
1,
2,
3,
4
],
[
1,
3,
5,
7
],
[
2,
3,
4,
5
],
[
3,
4,
5,
6
],
[
4,
5,
6,
7
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
1,
-1,
-1
]
],
"N": 9,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
1,
2,
3,
4
],
[
1,
3,
5,
7
],
[
2,
3,
4,
5
],
[
2,
4,
6,
8
],
[
3,
4,
5,
6
],
[
4,
5,
6,
7
],
[
5,
6,
7,
8
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
1,
-1,
-1
]
],
"N": 10,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
0,
3,
6,
9
],
[
1,
2,
3,
4
],
[
1,
3,
5,
7
],
[
2,
3,
4,
5
],
[
2,
4,
6,
8
],
[
3,
4,
5,
6
],
[
3,
5,
7,
9
],
[
4,
5,
6,
7
],
[
5,
6,
7,
8
],
[
6,
7,
8,
9
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
1,
-1,
-1
]
],
"N": 11,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
0,
3,
6,
9
],
[
1,
2,
3,
4
],
[
1,
3,
5,
7
],
[
1,
4,
7,
10
],
[
2,
3,
4,
5
],
[
2,
4,
6,
8
],
[
3,
4,
5,
6
],
[
3,
5,
7,
9
],
[
4,
5,
6,
7
],
[
4,
6,
8,
10
],
[
5,
6,
7,
8
],
[
6,
7,
8,
9
],
[
7,
8,
9,
10
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [
[
-1,
-1,
1,
1
],
[
-1,
1,
1,
-1
],
[
1,
-1,
-1,
1
],
[
1,
1,
-1,
-1
]
],
"N": 12,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
0,
3,
6,
9
],
[
1,
2,
3,
4
],
[
1,
3,
5,
7
],
[
1,
4,
7,
10
],
[
2,
3,
4,
5
],
[
2,
4,
6,
8
],
[
2,
5,
8,
11
],
[
3,
4,
5,
6
],
[
3,
5,
7,
9
],
[
4,
5,
6,
7
],
[
4,
6,
8,
10
],
[
5,
6,
7,
8
],
[
5,
7,
9,
11
],
[
6,
7,
8,
9
],
[
7,
8,
9,
10
],
[
8,
9,
10,
11
]
]
},
{
"route": "balanced four-block recurrence",
"seeds": [],
"N": 13,
"endpoint_progressions": [
[
0,
1,
2,
3
],
[
0,
2,
4,
6
],
[
0,
3,
6,
9
],
[
0,
4,
8,
12
],
[
1,
2,
3,
4
],
[
1,
3,
5,
7
],
[
1,
4,
7,
10
],
[
2,
3,
4,
5
],
[
2,
4,
6,
8
],
[
2,
5,
8,
11
],
[
3,
4,
5,
6
],
[
3,
5,
7,
9
],
[
3,
6,
9,
12
],
[
4,
5,
6,
7
],
[
4,
6,
8,
10
],
[
5,
6,
7,
8
],
[
5,
7,
9,
11
],
[
6,
7,
8,
9
],
[
6,
8,
10,
12
],
[
7,
8,
9,
10
],
[
8,
9,
10,
11
],
[
9,
10,
11,
12
]
]
}
]
}
],
"failures": []
}
Graph