Metrics of positive scalar curvature and unbounded widths

In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to $6$ and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar curvature metrics due to Gromov and Lawson. We develop analogies between the area of boundaries of special open subsets in our three-manifolds and $2$-colorings of associated full binary trees. Then, via combinatorial arguments and using the relative isoperimetric inequality, we argue that the widths converge to infinity.