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Source. Theorem 4, printed p. 1747, physical PDF p. 9. Harrington explicitly attributes this theorem to Chen; this page records Harrington's restatement rather than assigning a new proof to Harrington.

Conventions

For a positive integer $r$, define

$$Y_r=\left\{k>0:2\nmid k,\ k-2^n\text{ has at least }r \text{ distinct prime factors for every positive integer }n\right\},$$

and

$$G_r=\left\{k>0:2\nmid k,\ k2^n+1\text{ has at least }r \text{ distinct prime factors for every positive integer }n\right\}.$$

Statement

"If there exists a $(2,1)$-primitive $m$-covering system, then the sets $Y_{m+1}$ and $G_{m+1}$ each contain an infinite arithmetic progression." (p. 1747)

Harrington combines this restated implication with Theorem 2 to obtain Corollary 1: $Y_4$ and $G_4$ each contain an infinite arithmetic progression.

Proof scope. Harrington's paper states this as a result proved by Chen and does not reproduce its proof at this point. Chen's original paper was not checked for this card.