Learn German: Anhand der Farbe erkenne ich ein Gefäß tödlichen Giftes. - By means of the color, I recognize a vessel of deadly poison.

You can actually use either! The apposition ein Gefäß tödliches Gift (where the material simply mirrors the accusative case of the container) is very common and grammatically correct. However, using the Partitive Genitive (ein Gefäß tödlichen Giftes) creates a highly formal, literary, or dramatic tone—which fits perfectly when talking about a vessel of deadly poison.

Since there is no article before the material, shouldn't the strong adjective ending be -es instead of -en?

You would expect that, but this is a notorious exception! Normally, an adjective without an article takes the strong ending. However, for masculine and neuter nouns in the genitive, because the noun itself already takes an -s or -es ending (like Giftes), the adjective takes the weak -en ending instead to avoid repeating the "s" sound. Thus, it is tödlichen Giftes, never tödliches Giftes.

In everyday speech, would people actually use the Partitive Genitive like this?

Rarely. In spoken German, the Partitive Genitive sounds quite archaic or overly dramatic. Instead of ein Gefäß tödlichen Giftes, a native speaker in a normal conversation would usually use the preposition mit + dative (ein Gefäß mit tödlichem Gift) or the standard apposition (ein Gefäß tödliches Gift).

What case is der Farbe, and what triggers it?