We report new empirical and theoretical information about the dimension of trophic-niche space and the structure of food webs, as measured by the frequency of intervality and triangulation of overlap graphs and resource graphs in community food webs. Briand assembled and edited 113 community food webs, and informed us of which of those webs have interval trophic-niche overlap graphs. In this collection of 113 webs, the overall proportion of webs that are interval is as high as, or higher than, the proportion of interval webs observed previously. However, the fraction of webs that are interval is strongly associated with the number of species in the webs. The fraction of interval webs declines from one for small webs (16 or fewer species) toward zero for large webs (33 or more species). According to new mathematical and numerical calculations presented here, the cascade model predicts, as observed, that the probability that a web is interval is near one for webs with fewer than 10 species, declines as the ...