This is the story for the maximum size of a subset of [1,N]

            avoiding arithmetical progressions of size, 4, with gaps

                         less-than-or-equal to, 5,  .

                          ----------------------------

               Let F(N) be the largest size of a subset of [1,N]

avoiding arithmetical progressions of size , 4, with spacings <=, 1,  .  Then:

                              F(3 m + 1) = 4 m + 1

                              F(3 m + 2) = 4 m + 2

                              F(3 m + 3) = 4 m + 3

                              F(3 m + 4) = 4 m + 3

                      The asymptotic density is, 3/4,  .

                          ----------------------------

               Let F(N) be the largest size of a subset of [1,N]

avoiding arithmetical progressions of size , 4, with spacings <=, 2,  .  Then:

                              F(2 m + 1) = 3 m + 2

                              F(2 m + 2) = 3 m + 2

                              F(2 m + 3) = 3 m + 3

                      The asymptotic density is, 2/3,  .

                          ----------------------------

               Let F(N) be the largest size of a subset of [1,N]

avoiding arithmetical progressions of size , 4, with spacings <=, 3,  .  Then:

                             F(8 m + 1) = 12 m + 1

                             F(8 m + 2) = 12 m + 2

                             F(8 m + 3) = 12 m + 3

                             F(8 m + 4) = 12 m + 3

                             F(8 m + 5) = 12 m + 4

                             F(8 m + 6) = 12 m + 5

                             F(8 m + 7) = 12 m + 5

                             F(8 m + 8) = 12 m + 6

                             F(8 m + 9) = 12 m + 7

                             F(8 m + 10) = 12 m + 8

                             F(8 m + 11) = 12 m + 8

                             F(8 m + 12) = 12 m + 8

                      The asymptotic density is, 2/3,  .

                          ----------------------------

               Let F(N) be the largest size of a subset of [1,N]

avoiding arithmetical progressions of size , 4, with spacings <=, 4,  .  Then:

                              F(3 m + 1) = 5 m + 1

                              F(3 m + 2) = 5 m + 2

                              F(3 m + 3) = 5 m + 3

                              F(3 m + 4) = 5 m + 3

                              F(3 m + 5) = 5 m + 4

                      The asymptotic density is, 3/5,  .

                          ----------------------------

               Let F(N) be the largest size of a subset of [1,N]

avoiding arithmetical progressions of size , 4, with spacings <=, 5,  .  Then:

                              F(4 m + 1) = 7 m + 1

                              F(4 m + 2) = 7 m + 2

                              F(4 m + 3) = 7 m + 3

                              F(4 m + 4) = 7 m + 3

                              F(4 m + 5) = 7 m + 4

                              F(4 m + 6) = 7 m + 4

                              F(4 m + 7) = 7 m + 5

                      The asymptotic density is, 4/7,  .

                         This took, 1563.989, seconds