Positive partition relations for Pkappa,lambda

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Let kappa a regular uncountable cardinal and lambda a cardinal >kappa, and suppose lambda<kappa is less than the covering number for category cov(Mkappa,kappa). Then (a) Ikappa,lambda+ -->ᵏappa (Ikappa, lambda+,omega +1)², (b) Ikappa, lambda+ -->ᵏappa [Ikappa, lambda+]kappa+² if kappa is a limit cardinal, and (c) Ikappa, lambda+ -->ᵏappa (Ikappa, lambda+)² if kappa is weakly compact.