Erdős Problem #179 — Many short progressions force a longer progression

About 47 years old · traced to 1979

For integers $r<s$ , let $f_r(N;s)$ be the least number such that every sequence of $N$ integers containing at least $f_r(N;s)$ arithmetic progressions of length $r$ also contains one of length $s$ . Is $f_3(N;4)=o(N^2)$ ? More generally, is $f_3(N;s)=o(N^2)$ for every fixed $s$ ?

References

Primary source

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics, L'Enseignement Mathématique (2) 25 (1979), 325–344.

Additional references

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics, L'Enseignement Mathématique (2) 25 (1979), 325–344.

additive-combinatorics arithmetic-progressions

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