Observational constraints on star cluster formation theory - I. The
  mass-radius relation
  Stars form predominantly in groups usually denoted as clusters or
associations. The observed stellar groups display a broad spectrum of masses,
sizes and other properties, so it is often assumed that there is no underlying
structure in this diversity. Here we show that the assumption of an
unstructured multitude of cluster or association types might be misleading.
Current data compilations of clusters show correlations between cluster mass,
size, age, maximum stellar mass etc. In this first paper we take a closer look
at the correlation of cluster mass and radius. We use literature data to
explore relations in cluster and molecular core properties in the solar
neighborhood. We show that for embedded clusters in the solar neighborhood
there exists a clear correlation between cluster mass and half-mass radius of
the form $M_c = C R_c^{\gamma}$ with gamma = 1.7 +/-0.2. This correlation holds
for infra red K band data as well as X-ray sources and for clusters containing
a hundred stars up to those consisting of a few tens of thousands of stars. The
correlation is difficult to verify for clusters containing <30 stars due to
low-number statistics. Dense clumps of gas are the progenitors of the embedded
clusters. We find a similar slope for the mass-size relation of dense, massive
clumps as for the embedded star clusters. This might point at a direct
translation from gas to stellar mass: however, it is difficult to relate size
measurements for clusters (stars) to those for gas profiles. Taking into
account multiple paths for clump mass into cluster mass, we obtain an average
star-formation efficiency of 18%{+9.3}{-5.7} for the embedded clusters in the
solar neighborhood. The derived mass-radius relation gives constraints for the
theory of clustered star formation. Analytical models and simulations of
clustered star formation have to reproduce this relation in order to be
realistic (abridged)
