All spaces of countable spread can be small

The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{<\mathfrak{c}}=\mathfrak{c}$ with the following property () of partitions of pairs of $\mathfrak{c}$: \smallskip () For any coloring (or partition) $k:[\mathfrak{c}{]}^2\to 2$ either there is a homogeneous set of size $\mathfrak{c}$ in color $0$ or there is a set $S\in [\mathfrak{c}{]}^{\mathfrak{c}}$ such that for every countable $A\subset S$ there is $\beta \in \mathfrak{c}$ for which $A\subset \beta$ and $k(\{\alpha ,\beta \})=1$ for all $\alpha \in A$. \smallskip (*) plus $2^{<\mathfrak{c}}=\mathfrak{c}$ together then imply that for every topological space $X$ of countable spread, i.e. not containing any uncountable discrete subset, $\mathrm{\mid }X\mathrm{\mid }\le \mathfrak{c}$ if it is Hausdorff and $o(X)=\mathfrak{c}$ if it is also infinite and regular. Here $o(X)$ denotes the number of all open subsets of $X$.