Jensen proved that under $\Diamond$ there is a homogeneous Suslin continuum, so the square of a ccc homogeneous space can fail to be ccc. What about ccc topological groups?



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The obvious thing to try would be the free topological group over a Suslin continuum, but that doesn't work because it's a $\sigma$-compact group and, by a result of Tkachenko, $\sigma$-compact groups are ccc (and the square of a $\sigma$-compact group is clearly a $\sigma$-compact group).



The square of a ccc topological group
Yes. There are such groups after adding a Cohen real (see Theorem $4$ in "Nonpreservation of properties of topological groups on taking their square", Malykhin, 1987) or under RVM (see Theorem $0^c$ in "Some applications of S and L combinatorics", Todorcevic, 1993).


