Induced Turán problems: Largest Pm-free graphs with bounded degree

A graph is H-free if it has no induced subgraph isomorphic to H. Let F(D; H) be the family of H-free connected graphs with maximum degree at most D. Let ex∗(D; H) = max{|E(G)| : G ∈ F(D; H)}; this is finite only when H is a disjoint union of paths. In that case, ex∗(D; H) ∈ O(D2 ex∗(D; Pm)), where Pm is the longest path in H. For m ≥ 6, we determine ex∗(D; Pm) asymptotically (with m fixed), showing that ex∗(D; Pm) ∼ 1 2D for even m and ex∗(D; Pm) ∼ 18D for odd m. We also compute max{|V (G)| : G ∈ F(D; Pm)} when m is odd. Finally, we compute ex∗(D; P5) exactly for D ≥ 187: for example, it equals 2 27D + 7 18D + 12D when D ≡ 0 mod 3.