Exercise 6, § 2

Let $k\geqslant 1$ be an integer. For every integer $N\geqslant 1$, denote by $S_k(N)$ the largest cardinal of a set $A\subset \{1, . . . ,N\}$ which does not contain $k$ elements in arithmetic progression, that is to say that, whatever $n_0\geqslant 0,h\geqslant 1$, $\ell \geqslant 1$, if $\{n_0+h, n_0+ 2h, . . . , n_0+\ell h\} \subset A$, then $\ell < k$.

a) Now prove that the limit

$s_k=$ lim$_{N\rightarrow+\infty}\frac{S_k(N)}{N}$

exists. b) Now prove that $s_1=s_2= 0$. It can be shown (« Szemerédi theorem », cf. E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arithmetica XXVII, 1975, 199–245) that $s_k= 0$ for every $k\geqslant 1$ ; for the case $k= 3$, cf. exercise 23 of II, p. 270.