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An arithmetic progression is a (possibly infinite) sequence of numbers such that the difference between consecutive terms is always the same value.
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Largest number of k-arithmetic progressions without a (k+1)-arithmetic progression
Suppose $A \subseteq \{1,\dots,n\}$ does not contain any arithmetic progressions of length $k+1$. What is the largest number of $k$-term arithmetic progressions that $A$ can have? (one may also wish t …