Counting flags in triangle-free digraphs. Jan Hladký (Charles Uni, Prague & TU Munich) Daniel Král (Charles Uni, Prague) Sergey Norin (Princeton Uni)

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1 Counting flags in triangle-free digraphs Jan Hladký (Charles Uni, Prague & TU Munich) Daniel Král (Charles Uni, Prague) Sergey Norin (Princeton Uni)

2 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D)

3 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r.

4 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle.

5 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle. Theorem There is no triangle-free digraph D on n vertices with δ + (D) cn.

6 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle. Theorem There is no triangle-free digraph D on n vertices with δ + (D) cn. Caccetta, Häggkvist, 1978: c = ,

7 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle. Theorem There is no triangle-free digraph D on n vertices with δ + (D) cn. Caccetta, Häggkvist, 1978: c = , Bondy, 1997: c = ,

8 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle. Theorem There is no triangle-free digraph D on n vertices with δ + (D) cn. Caccetta, Häggkvist, 1978: c = , Bondy, 1997: c = , Shen, 1998: c = ,

9 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle. Theorem There is no triangle-free digraph D on n vertices with δ + (D) cn. Caccetta, Häggkvist, 1978: c = , Bondy, 1997: c = , Shen, 1998: c = , Hamburger, Haxell, Kostochka, 2007: c = ,

10 digraph...no loops, no parallel edges, no counterparallel edges minimum outdegree of a digraph D...δ + (D) Any digraph D on n vertices with δ + (D) n/r contains a (directed) cycle of length at most r. Any digraph D on n vertices with δ + (D) n/3 contains a triangle. Theorem There is no triangle-free digraph D on n vertices with δ + (D) cn. Caccetta, Häggkvist, 1978: c = , Bondy, 1997: c = , Shen, 1998: c = , Hamburger, Haxell, Kostochka, 2007: c = , HKN, 2009: c =

11 Razborov s theory of Flag Algebras limits of graph...graphons

12 Razborov s theory of Flag Algebras limits of graph...graphons, graphon W : G [0, 1]

13 Razborov s theory of Flag Algebras limits of graph...graphons, graphon W : G [0, 1]

14 Razborov s theory of Flag Algebras limits of graph...graphons, graphon W : G [0, 1]

15 Razborov s theory of Flag Algebras limits of graph...graphons, graphon W : G [0, 1] Averaging:

16 Razborov s theory of Flag Algebras limits of graph...graphons, graphon W : G [0, 1] Averaging: Asymptotic version of Mantel s Theorem Suppose that W is a graphon with W (K 2 ) > 1/2. Then W (K 3 ) > 0.

17 Our proof Observation Suppose that D is a triangle-free digraph on n vertices with δ + (D) cn. Then for any m 0 there exist m > m 0 and a triangle-free digraph D on m vertices with δ + (D ) cm.

18 Our proof Observation Suppose that D is a triangle-free digraph on n vertices with δ + (D) cn. Then for any m 0 there exist m > m 0 and a triangle-free digraph D on m vertices with δ + (D ) cm. Main Theorem Suppose that W is a triangle-free digraphon. Then δ + (W ) <

19 Our proof Observation Suppose that D is a triangle-free digraph on n vertices with δ + (D) cn. Then for any m 0 there exist m > m 0 and a triangle-free digraph D on m vertices with δ + (D ) cm. Main Theorem Suppose that W is a triangle-free digraphon. Then δ + (W ) < Proof Take W to be a triangle-free digraphon attaining maximum value δ + (W ) and suppose for contradiction that δ + (W ) Ingredients: Chudnovsky-Seymour-Sullivan, 2008: Every triangle-free digraph with k nonedges can be made acyclic by deleting at most k edges. improved recently by Dunkum, Hamburger, Pór Cauchy-Schwarz Inequality. Induction.