Edge inducibility grows as a power of edge count

What the study found: The paper shows that, for any graph H, the maximum number of induced copies of H in an m-edge graph is Theta(m^alpha_f(H)), where alpha_f(H) is the fractional independence number of H.
Why the authors say this matters: The authors use this result to focus attention on the constant factor in front of m^alpha_f(H), and they say they give additional results for paths and cycles.
What the researchers tested: The study examines the edge version of inducibility, defining rho(H,m) as the maximum number of induced copies of H among graphs with exactly m edges. The authors use the entropy method to prove their general theorem and to derive bounds and conjectures for specific graphs.
What worked and what didn't: For cycles C_k with k >= 5, the authors conjecture rho(C_k,m) = (1+o(1))(m/k)^(k/2), with the bound achieved by the blow up of C_k. They establish an upper bound with an extra constant factor for even cycles, and for odd cycles they obtain an upper bound with an extra factor depending on k. For paths, they prove rho(P_{2l},m) <= m^l / [2(l-1)^(l-1)] and rho(P_{2l+1},m) <= m^(l+1) / [4l^l] for l >= 2, and they also conjecture the asymptotic value of rho(P_k,m).
What to keep in mind: The abstract does not provide proofs in detail, and some cycle and path statements are conjectures rather than established equalities. The summary is limited to the cases and bounds stated in the abstract.

Key points

• For any graph H, rho(H,m) is Theta(m^alpha_f(H)).
• The paper studies the edge version of inducibility, where graphs are constrained by having exactly m edges.
• For cycles C_k with k >= 5, the authors conjecture an asymptotic formula and say it is achieved by the blow up of C_k.
• The authors prove upper bounds for even and odd cycles, with an extra factor depending on k for odd cycles.
• For paths P_{2l} and P_{2l+1}, the paper gives explicit upper bounds and also conjectures the asymptotic value of rho(P_k,m).