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A<sc>bstract</sc> An extended search for anomaly free matter coupledN= (1,0) supergravity in six dimension is carried out by two different methods which we refer to as the graphical and rank methods. In the graphical method the anomaly free models are built from single gauge group models, called nodes, which can only have gravitational anomalies. We search for anomaly free theories with gauge groupsG₁× … ×G_nwithn= 1,2,… (any number of factors) andG₁× … ×G_n×U(1)_Rwheren= 1,2,3 andU(1)_Ris theR-symmetry group. While we primarily consider models with the tensor multiplet numbern_T= 1, we also provide some results forn_T≠ 1 with an unconstrained number of charged hypermultiplets. We find a large number of ungauged anomaly free theories. However, in the case ofR-symmetry gauged models withn_T= 1, in addition to the three known anomaly free theories withG₁×G₂×U(1)_Rtype symmetry, we find only six new remarkably anomaly free models with symmetry groups of the formG₁×G₂×G₃×U(1)_R. In the case ofn_T= 1 and ungauged models, excluding low rank group factors and considering only low lying representations, we find all anomaly free theories. Remarkably, the number of group factors does not exceed four in this class. The proof of completeness in this case relies on a bound which we establish for a parameter characterizing the difference between the number of non-singlet hypermultiplets and the dimension of the gauge group.