Suppose $P$ and $Q$ are ccc partial orders. Is $P \times Q$ $\omega\_2$-cc? Note that this true under CH by the Erdos-Rado Theorem.



chain condition of a product of posets
First of all, note (as Monroe does in his question) that if $\mathbb P,\mathbb Q$ are ccc, then $\mathbb P\times\mathbb Q$ is $\mathfrak c^+$-cc, as an immediate consequence of the Erdős-Rado theorem $(2^{\aleph\_0})^+\to(\aleph\_1)^2\_2$. (This is to say, if $\mathbb P$ and $\mathbb Q$ do not admit uncountable antichains, then any antichain in their product has size at most continuum.) The question is essentially whether we can do better.


Under Martin's axiom ($\mathsf{MA}\_{\aleph\_1}$ suffices) we can: $\mathsf{MA}\_{\aleph\_1}$ implies that any ccc poset has property (K), that is, not only it does not have uncountable antichains, but in fact any uncountable set contains an uncountable subset of pairwise compatible conditions. The product of two property (K) posets has property (K) as well (and, more strongly, so does the finite support product of any number of property (K) posets). Thus, under $\mathsf{MA}\_{\aleph\_1}$, the product of two ccc posets is again ccc.


Without such an assumption, the situation can be completely different. For instance, if $\mathbb S$ is a Suslin tree, then it is ccc but its square is not. The failure of the ccc is not too bad in this case: Since $|\mathbb S|=\aleph\_1$, $\mathbb S\times\mathbb S$ is $\aleph\_2$-cc. From this, the question presents itself, as it is perhaps natural to wonder whether we can prove that a product of two ccc posets is always $\aleph\_2$-cc.


This cannot be done in general: It is consistent with the continuum as large as desired that there are ccc posets whose product admits an antichain of size continuum (that is, the trivial upper bound is best possible). This was first proved in 



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> [MR0493930 (58 #12886)](http://www.ams.org/mathscinet-getitem?mr=493930). Fleissner, William G. *[Some spaces related to topological inequalities proven by the Erdős-Rado theorem](http://www.ams.org/journals/proc/1978-071-02/S0002-9939-1978-0493930-4/S0002-9939-1978-0493930-4.pdf)*. Proc. Amer. Math. Soc. **71** (1978), no. 2, 313–320. 
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His model is obtained simply by adding $\kappa$ Cohen reals, for $\kappa$ an uncountable cardinal. The argument is elegant, the two posets are defined in terms of a coloring $P:[\kappa]^2\to 2$: $\mathbb P$ is defined by considering the maximal 0-homogeneous subsets of $\kappa$, and $\mathbb Q$ by considering the maximal 1-homogeneous subsets. (Actually, Fleissner describes topologies on $\mathbb P,\mathbb Q$ and shows that the cellularity of the product is at least $\kappa$.) The difficulty is finding $P$ that ensures $\mathbb P,\mathbb Q$ are ccc to begin with. It is here that forcing is used, with $P$ added generically. See section 5 in Fleissner's paper for details. 


The topic of cellularity of products is quite interesting and there are many additional results. An excellent survey can be found in



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> [MR3271280](http://www.ams.org/mathscinet-getitem?mr=3271280). Rinot, Assaf. *[Chain conditions of products, and weakly compact cardinals](https://doi.org/10.1017/bsl.2014.24)*. Bull. Symb. Log. **20** (2014), no. 3, 293–314. 
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(Assaf's paper is also [available](http://blog.assafrinot.com/?p=3755) at his page.)


This situation, of having two ccc posets whose product is not $\mathfrak c$-cc, can also be arranged in a variety of other ways. For example, it is a consequence of the continuum being real-valued measurable. For this, see for instance Theorem 7D in 



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> [MR1234282 (95f:03084)](http://www.ams.org/mathscinet-getitem?mr=1234282). Fremlin, D. H. *[Real-valued-measurable cardinals](https://www1.essex.ac.uk/maths/people/fremlin/rvmc.pdf)*. In **Set theory of the reals** (Ramat Gan, 1991), 151–304, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993. 
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