We isolate a new class of forcing notions intermediate between \(\sigma\)-centered and c.c.c.
This class is closed with respect to regular subposets and finite support iteration, it is not productive nor productively c.c.c.
Forcings in this class add unbounded reals, do not add random reals, cofinal branches into \(\omega_1\)-trees, uncountable cliques in closed graphs, nor separate uncountable gaps.
Most posets, where the proof of c.c.c. $\,$ uses a straightforward \(\Delta\)-system argument, fall into this class.
This forcing property appeared implicitly in recent work of T. Yorioka.
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