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An extended search for anomaly free matter coupled $N=(1,0)$ supergravity insix dimension is carried out by two different methods which we refer to as thegraphical and rank methods. In the graphical method the anomaly free models arebuilt from single gauge group models, called nodes, which can only havegravitational anomalies. We search for anomaly free theories with gauge groups$G_1\times...\times G_n$ with $n=1,2,...$ (any number of factors) and$G_1\times...\times G_n \times U(1)_R$ where $n=1,2,3$ and $U(1)_R$ is the$R$-symmetry group. While we primarily consider models with the tensormultiplet number $n_T=1$, we also provide some results for $n_T\ne 1$ with anunconstrained number of charged hypermultiplets. We find a large number ofungauged anomaly free theories. However, in the case of $R$-symmetry gaugedmodels with $n_T=1$, in addition to the three known anomaly free theories with$G_1\times G_2 \times U(1)_R$ type symmetry, we find only six new remarkablyanomaly free models with symmetry groups of the form $G_1\times G_2\times G_3\times U(1)_R$. In the case of $n_T=1$ and ungauged models, excluding low rankgroup factors and considering only low lying representations, we find allanomaly free theories. Remarkably, the number of group factors does not exceedfour in this class. The proof of completeness in this case relies on a boundwhich we establish for a parameter characterizing the difference between thenumber of non-singlet hypermultiplets and the dimension of the gauge group.

New anomaly free supergravities in six dimensions