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Given a positive whole number n, [tex]\exists[/tex] N with the following property: if A is a subgroup of {1,2,...,N} with at least N/2 elements, then there is a positive whole number m<= N - n such that

|A [tex]\cap[/tex]{m+1, m+2,..., m+k}|>=k/2

[tex]\forall[/tex] k = 1, 2, …, n.

|A [tex]\cap[/tex]{m+1, m+2,..., m+k}|>=k/2

[tex]\forall[/tex] k = 1, 2, …, n.

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