I have recently been reading Terence Tao's paper on the Szemeredi's original proof of Szemeredi's theorem (https://terrytao.files.wordpress.com/2017/09/szemeredi-proof1.pdf) and I have come upon this part of the proof of the theorem 1.6 (kind of a key result before proving Roth's theorem) which I fail to understand.
More precisely, it is stated that in (5.5) there is some $l'\in[L']$ such that $|\{h'\in E_h:U(P_{l'}(2)+h')\in A\}|>0$ for all but at most $\frac{1}{100L}H$ of the $h\in W$, where $W$ is a subset of $[H]$ such that $|W|\geq\frac{1}{30}H$ (claim (5.3). Then it is stated in (5.6) that $U(P_{l'}(3)+h)\in S$ for all but at most $\frac{1}{9L}H$ of the $h\in[H]$.
Now comes the tricky part. It is claimed that from these statements one can derive that $W$ must contain an arithmetic progression of the length $L$ and of the form $h_0+2\overrightarrow{[L]}$ whose elements satisfy (5.5) and (5.6). This would however mean that one can find $L$ consecutive elements of $W$ satisfying these two conditions, since $W$ is defined to contain numbers of the same parity. The issue here is that by (5.5) and (5.6) it is possible that $(\frac{1}{100L}H+\frac{1}{9L})H>\frac{1}{30L}H$ of the $h\in W$ are "bad", even though by the pigeonhole principle one can have at most $|W|/L\geq\frac{1}{30L}H$ "bad" elements in $W$ and be sure that there are $L$ consecutive "good" ones.
I tried to reread the proof to see if I'm missing some points apart from (5.3), (5.5), and (5.6) that are used to derive the previous claim, but it seems to me that it should be obtained directly from these three.