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

            avoiding arithmetical progressions of size, 3, with gaps

                         less-than-or-equal to, 7,  .

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

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

avoiding arithmetical progressions of size , 3, with spacings <=, 1,  .  Then:

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

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

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

                      The asymptotic density is, 2/3,  .

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

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

avoiding arithmetical progressions of size , 3, with spacings <=, 2,  .  Then:

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

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

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

                      The asymptotic density is, 2/3,  .

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

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

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

                              F(4 m + 1) = 8 m + 1

                              F(4 m + 2) = 8 m + 2

                              F(4 m + 3) = 8 m + 2

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

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

                              F(4 m + 6) = 8 m + 4

                              F(4 m + 7) = 8 m + 4

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

                      The asymptotic density is, 1/2,  .

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

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

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

                              F(4 m + 1) = 9 m + 1

                              F(4 m + 2) = 9 m + 2

                              F(4 m + 3) = 9 m + 2

                              F(4 m + 4) = 9 m + 3

                              F(4 m + 5) = 9 m + 4

                              F(4 m + 6) = 9 m + 4

                              F(4 m + 7) = 9 m + 4

                              F(4 m + 8) = 9 m + 4

                              F(4 m + 9) = 9 m + 5

                      The asymptotic density is, 4/9,  .

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

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

avoiding arithmetical progressions of size , 3, with spacings <=, 5,  .  Then:

                              F(4 m + 1) = 9 m + 1

                              F(4 m + 2) = 9 m + 2

                              F(4 m + 3) = 9 m + 2

                              F(4 m + 4) = 9 m + 3

                              F(4 m + 5) = 9 m + 4

                              F(4 m + 6) = 9 m + 4

                              F(4 m + 7) = 9 m + 4

                              F(4 m + 8) = 9 m + 4

                              F(4 m + 9) = 9 m + 5

                      The asymptotic density is, 4/9,  .

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

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

avoiding arithmetical progressions of size , 3, with spacings <=, 6,  .  Then:

                              F(4 m + 1) = 9 m + 1

                              F(4 m + 2) = 9 m + 2

                              F(4 m + 3) = 9 m + 2

                              F(4 m + 4) = 9 m + 3

                              F(4 m + 5) = 9 m + 4

                              F(4 m + 6) = 9 m + 4

                              F(4 m + 7) = 9 m + 4

                              F(4 m + 8) = 9 m + 4

                              F(4 m + 9) = 9 m + 5

                      The asymptotic density is, 4/9,  .

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

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

avoiding arithmetical progressions of size , 3, with spacings <=, 7,  .  Then:

                              F(4 m + 1) = 9 m + 1

                              F(4 m + 2) = 9 m + 2

                              F(4 m + 3) = 9 m + 2

                              F(4 m + 4) = 9 m + 3

                              F(4 m + 5) = 9 m + 4

                              F(4 m + 6) = 9 m + 4

                              F(4 m + 7) = 9 m + 4

                              F(4 m + 8) = 9 m + 4

                              F(4 m + 9) = 9 m + 5

                      The asymptotic density is, 4/9,  .

                          This took, 894.413, seconds