## Maximal lines free sets.

These data were generated for the Polymath1-project.

#### Combinatorial lines

d is the number of dimensions and k the number of colours.

An entry A/B menas that an optimal solution has size A and there are B optimal solutions.

d\k | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 2/1 | 6/4 | 12/15 | 20/76 | 30/455 | 42/3168 |

3 | 3/2 | 18/1 | 48/2 | 100/198 | 180/11616 | 294/? |

4 | 6/1 | 52/3 | 183/32 | 500/60 | 1051-1079/? | 2058/? |

5 | 10/2 | 150/12 | 712-762/? | 2500/6 |

Here are the optimal solutions, and largest examples found so far for some cases.

These are the maximum sizes of a set without combinatorial lines where the set consist of all point with a given coordinate sum.
d\k | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

4 | 6/1 | 52/1 | 179/2 | 500/5 | 1042/4 | 2058/7 | 3516/4 | 5776/2 | 8895/2 |

5 | 10/2 | 150/1 | 712/1 | 2500/1 | 6325/4 | 14406/7 | 28101/4 | 51557/2 | 88669/2 |

6 | 20/1 | 432/1 | 2807/2 | 12486/1 | 37372/2 | 100842/7 | ? | ? | ? |

7 | 35/7 | 1248/1 | 11091/2 | 62260/1 | ? | ? | ? | ? | ? |

8 | 70/1 | 3618/1 | 43515/2 | ? | ? | ? | ? | ? | ? |

9 | 126/9 | 10530/1 | ? | ? | ? | ? | ? | ? | ? |

Here are the optimal solutions, given as lists of the coordinate sums for the included vertices.

## Optimal sets for Fujimura's problem.

Fujimura's problem is described here.

n is the sum and k the size of the tuples.

An entry A/B menas that an optimal solution has size A and there are B optimal solutions.

n\k | 3 | 4 | 5 | 6 |

2 | 4/ | 7/30 | 12/15 | 17/315 |

3 | 6/10 | 14/78 | 28/30 | 45/118764 |

4 | 9/1 | 24/224 | 56/30 | 101/? |

5 | 12/1 | 37/21296 | 100/126 | 202-205/ |

6 | 15/4 | 55/33152 | 164/18750 | / |

7 | 18/85 | 78/48 | ?/? | / |

8 | 22/72 | ?/? | ?/? | / |

9 | 26/183 | ?/? | ?/? | / |

10 | 31/6 | ?/? | ?/? | / |

11 | 35/576 | ?/? | ?/? | / |

12 | 40/876 | ?/? | ?/? | / |

13 | 46/54 | ?/? | ?/? | / |

Here are the optimal solutions.

## Optimal sets for the weighted Fujimura problem corresponding to combinatorial lines.

n is the sum and k the size of the tuples.

An entry A/B menas that an optimal solution has weigth A and there are B optimal solutions.

n\k | 3 |

3 | 18/1 |

4 | 52/3 |

5 | 150/12 |

6 | 450/1 |

7 | 1302/12 |

8 | 3780/12 |

9 | 11340/1 |

10 | 32864/12 |

11 | 96338/? |

12 | 287892/? |

13 | 854139/? |

14 | 2537821/? |

15 | 7528835/? |

16 | 22517082/? |

17 | 66944301/? |

18 | 198629224/? |

19 | 593911730/? |

20 | 1766894722/? |

Here are the optimal solutions.

## Optimal sets for the Fujimura type problem for Moser sets.

n is the sum and k the size of the tuples.

An entry A/B menas that an optimal solution has weigth A and there are B optimal solutions.

n\k | 3 |

3 | 16/2 |

4 | 43/2 |

5 | 122/5 |

6 | 353/2 |

7 | 1017/2 |

8 | 2902/2 |

9 | 8622/1 |

10 | 24786/2 |

11 | 71766/2 |

12 | 212423/2 |

13 | 614875/2 |

14 | 1794212/1 |

15 | 5321796/2 |

16 | 15455256/1 |

17 | 45345052/? |

18 | 134438520/? |

19 | 391796798/? |

20 | 1153402148/? |

Here are the optimal solutions.

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