  Inference in clustering is paramount to uncovering inherent group structure
in data. Clustering methods which assess statistical significance have recently
drawn attention owing to their importance for the identification of patterns in
high dimensional data with applications in many scientific fields. We present
here a U-statistics based approach, specially tailored for high-dimensional
data, that clusters the data into three groups while assessing the significance
of such partitions. Because our approach stands on the U-statistics based
clustering framework of the methods in R package uclust, it inherits its
characteristics being a non-parametric method relying on very few assumptions
about the data, and thus can be applied to a wide range of dataset. Furthermore
our method aims to be a more powerful tool to find the best partitions of the
data into three groups when that particular structure is present. In order to
do so, we first propose an extension of the test U-statistic and develop its
asymptotic theory. Additionally we propose a ternary non-nested significance
clustering method. Our approach is tested through multiple simulations and
found to have more statistical power than competing alternatives in all
scenarios considered. Applications to peripheral blood mononuclear cells and to
image recognition shows the versatility of our proposal, presenting a superior
performance when compared with other approaches.
