For every k ≥ 3 there exist k-graphs F1 and F2 with 0 < γ⁺(F1, F2) < min{γ⁺(F1), γ⁺(F2)}.
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Existence of linear k-graphs with codegree Turán density arbitrarily close to zero is proved via affine-plane incidence structures over finite fields.
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Strong non-principality of positive codegree Tur\'an density
For every k ≥ 3 there exist k-graphs F1 and F2 with 0 < γ⁺(F1, F2) < min{γ⁺(F1), γ⁺(F2)}.
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On linear $k$-graphs with codegree Tur\'an density arbitrarily close to zero
Existence of linear k-graphs with codegree Turán density arbitrarily close to zero is proved via affine-plane incidence structures over finite fields.