论文信息 - Integer Sets Containing No Solution to x + y = 3z

Integer Sets Containing No Solution to x + y = 3z

We prove that a maximum subset of {1,2, …, n} containing no solutions to x + y = 3z has $$ \left\lceil {\frac{n}{2}} \right\rceil $$ elements if n ≠ 4, thus settling a conjecture of Erdos. For n ≥ 23 the set of all odd integers less than or equal to n is the unique maximum such subset.

Fan Chung Graham | John L. Goldwasser

[1] F. Behrend. On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.

[2] Fan Chung Graham,et al. Maximum subsets of (0, 1] with no solutions to x+y = kz , 1996, Electron. J. Comb..

[3] R. Salem,et al. On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[4] E. Szemerédi. On sets of integers containing k elements in arithmetic progression , 1975 .

[5] K. F. Roth. On Certain Sets of Integers , 1953 .

[6] D. R. Heath-Brown. Integer Sets Containing No Arithmetic Progressions , 1987 .

[7] E. Szemerédi. On sets of integers containing no four elements in arithmetic progression , 1969 .