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editing
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a(9) = 1 is the well-known fact that the Van der Waerden number for two colors and three-term arithmetic progressions is 9.
proposed
editing
a(9)=1 is the well known fact that the Van der Waerden number for 2 colors and three-term arithmetic progressions is 9.
proposed
editing
a(9)=1 is the well known fact that the van der Waerden number for 2 colors and three-term arithmetic progressions is 9.
The general problem for k terms and r colors can be solved by using integer programming, with binary decision variables x[i,c] = 1 if element i receives color c and 0 otherwise, y[i,d] = 1 if arithmetic progression (i,i+d,...,i+(k-1)d) is monochromatic and 0 otherwise. The constraints are sum {c in 1..r} x[i,c] = 1 for all i in 1,…,n and sum {j=0 to k-1} x[i+j*d,c] - k + 1 <= y[i,d] for all i, d, c. The objective is to minimize sum {i, d} y[i,d].
a(8)=0 because we can two color {1,...,8} by 11001100 so that there are no monochromatic three-term arithmetic progressions.
proposed
editing
The general problem for k terms and r colors can solved by using integer programming, with binary decision variables x[i,c] = 1 if element i receives color c and 0 otherwise, y[i,d] = 1 if arithmetic progression (i,i+d,...,i+(k-1)d) is monochromatic and 0 otherwise. The constraints are sum {c in 1..r} x[i,c] = 1 for all i in 1,…,n and sum {j=0 to k-1} x[i+j*d,c] - k + 1 <= y[i,d] for all i, d, c. The objective is to minimize sum {i, d} y[i,d].
a(8)=0 because we can two color {1,...,8} by 11001100 so that there are no three-term arithmetic progressions.
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reviewed
proposed