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.
Scobee, On the Ramsey numberr(m 1P3, m2P3, m3P3) and related results, M.A
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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.