The conjectured formula for R(n1P3,...,nkP3) is false in general but holds under some conditions; the Gallai-Ramsey version GR(n1P3,...,nkP3) equals the formula.
Irving, Generalised Ramsey numbers for small graphs,Discrete Math.9(1974), 251–264
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Ramsey numbers and Gallai--Ramsey numbers of disjoint unions of cherries
The conjectured formula for R(n1P3,...,nkP3) is false in general but holds under some conditions; the Gallai-Ramsey version GR(n1P3,...,nkP3) equals the formula.