In this paper, we prove several new Turán density results for 3-graphs with independent neighbourhoods. We show:\begin{align*} \pi (K_4^-, C_5, F_{3,2})=12/49, \pi (K_4^-, F_{3,2})=5/18 \textrm {and} \pi (J_4, F_{3,2})=\pi (J_5, F_{3,2})=3/8, \end{align*}whereJtis the 3-graph consisting of a single vertexxtogether with a disjoint setAof sizetand all$\binom{|A|}{2}$3-edges containingx. We also prove two Turán density results where we forbid certain induced subgraphs:\begin{align*} \pi (F_{3,2}, \textrm { induced }K_4^-)=3/8 \textrm {and} \pi (K_5, 5\textrm {-set spanning exactly 8 edges})=3/4. \end{align*}The latter result is an analogue forK5of Razborov's result that\begin{align*} \pi (K_4, 4\textrm {-set spanning exactly 1 edge})=5/9. \end{align*}We give several new constructions, conjectures and bounds for Turán densities of 3-graphs which should be of interest to researchers in the area. Our main tool is ‘Flagmatic’, an implementation of Razborov's semi-definite method, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the semi-definite method. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the ‘complexity barrier’ for the semi-definite method may be of general interest.
Erdős asked in 1962 about the value of f (n, k, l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k, l) = (3, 3) was solved by Lorden. Here we solve the problem (for all large n) for (3, l) with 4 ≤ l ≤ 7 and (k, 3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves, and Sudakov resolved the cases (k, l) = (3, 4) and (4, 3).
Given an r-graph H on h vertices, and a family F of forbidden subgraphs, we define ex H (n, F ) to be the maximum number of induced copies of H in an F -free r-graph on n vertices. Then the Turán H-density of F is the limit π H (F ) = lim n→∞
Abstract. Given a family of 3-graphs F , we define its codegree threshold coex(n, F ) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F 3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe, and cde. In this paper, we give two proofs that coex(n, {F 3,2 }) = + o(1) n, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large n: coex(n, {F 3,2 }) = n/3 − 1 if n is congruent to 1 modulo 3, and n/3 otherwise. In addition we determine the set of codegree-extremal configurations for all sufficiently large n.
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