The amalgamation decision problem asks whether the finite models of a given universal first-order sentence have the amalgamation property. We prove that this problem is ExpTime-hard for universal Horn sentences of maximum relation arity three, and 2ExpTime-hard when the arity is unbounded, thereby strengthening the previous best lower bounds for universal Horn sentences. The first reduction simulates runs of nondeterministic tree automata using a single ternary relation and unary and binary auxiliary relations. The second simulates finite winning strategies in an exponential corridor game. As a secondary contribution, we prove that deciding the winner remains 2ExpTime-complete for the game normal form used in our reduction, in which each move label determines a unique legal successor from every non-winning legal row.

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