Edges and Triangles. Po-Shen Loh. Carnegie Mellon University. Joint work with Jacob Fox

Transcription

1 Edges and Triangles Po-Shen Loh Carnegie Mellon University Joint work with Jacob Fox

2 Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no edge is in more than one triangle.

3 Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no edge is in more than one triangle.

4 Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no edge is in more than one triangle. Question (Erdős-Rothschild) What if the total number of edges must be at least 0.001n? Must some edge be in many triangles?

5 Regularity Lemma Szemerédi Regularity Lemma For every ɛ, there is M such that every graph can be ɛ-approximated by an object of complexity bounded by M. Triangle Removal Lemma For any ɛ, there is a δ such that every graph with δn 3 triangles can be made triangle-free by deleting only ɛn edges.

6 Regularity Lemma Szemerédi Regularity Lemma For every ɛ, there is M such that every graph can be ɛ-approximated by an object of complexity bounded by M. Triangle Removal Lemma For any ɛ, there is a δ such that every graph with δn 3 triangles can be made triangle-free by deleting only ɛn edges. Dependency between parameters: (From Regularity Lemma.) 1 δ is tower of height power of 1 ɛ. (Fox.) 1 δ is tower of height logarithmic in 1 ɛ.

7 Lower bound for Erdős-Rothschild Observation Let c be a constant. Given cn edges, each of which is in a triangle, there is always an edge which is in log n triangles.

8 Lower bound for Erdős-Rothschild Observation Let c be a constant. Given cn edges, each of which is in a triangle, there is always an edge which is in log n triangles. Proof: If the graph has over δn 3 triangles, then double-counting already gives an edge in at least 3δn3 triangles. cn

9 Lower bound for Erdős-Rothschild Observation Let c be a constant. Given cn edges, each of which is in a triangle, there is always an edge which is in log n triangles. Proof: If the graph has over δn 3 triangles, then double-counting already gives an edge in at least 3δn3 triangles. cn Else, Removal Lemma gives ɛn edges hitting all the triangles.

10 Lower bound for Erdős-Rothschild Observation Let c be a constant. Given cn edges, each of which is in a triangle, there is always an edge which is in log n triangles. Proof: If the graph has over δn 3 triangles, then double-counting already gives an edge in at least 3δn3 triangles. cn Else, Removal Lemma gives ɛn edges hitting all the triangles. Every edge is in a triangle, so total number of triangles cn 3. Then some edge is in at least cn /3 ɛn triangles.

11 Lower bound for Erdős-Rothschild Observation Let c be a constant. Given cn edges, each of which is in a triangle, there is always an edge which is in log n triangles. Proof: If the graph has over δn 3 triangles, then double-counting already gives an edge in at least 3δn3 triangles. cn Else, Removal Lemma gives ɛn edges hitting all the triangles. Every edge is in a triangle, so total number of triangles cn 3. Then some edge is in at least cn /3 triangles. ɛn Either case gives an edge in at least min{ 3δn c, c 3ɛ } triangles.

12 Lower bound for Erdős-Rothschild Observation Let c be a constant. Given cn edges, each of which is in a triangle, there is always an edge which is in log n triangles. Proof: If the graph has over δn 3 triangles, then double-counting already gives an edge in at least 3δn3 triangles. cn Else, Removal Lemma gives ɛn edges hitting all the triangles. Every edge is in a triangle, so total number of triangles cn 3. Then some edge is in at least cn /3 triangles. ɛn Either case gives an edge in at least min{ 3δn c, c 3ɛ } triangles. Take 1 δ = n, and 1 ɛ = power of log n.

13 Previous work Theorem (Alon-Trotter) For any constant c < 1 4, there is a cn -edge graph with every edge in a triangle, but the most popular edge only in n triangles.

14 Previous work Theorem (Alon-Trotter) For any constant c < 1 4, there is a cn -edge graph with every edge in a triangle, but the most popular edge only in n triangles. Theorem (Edwards; Khadžiivanov-Nikiforov) Given any 1 4 n edges, there is always one in 1 6 n triangles.

15 Previous work Theorem (Alon-Trotter) For any constant c < 1 4, there is a cn -edge graph with every edge in a triangle, but the most popular edge only in n triangles. Theorem (Edwards; Khadžiivanov-Nikiforov) Given any 1 4 n edges, there is always one in 1 6 n triangles. Theorem (Bollobás-Nikiforov) Given any 1 4 n o(n 1.4 ) edges, each of which is in a triangle, there is always some edge in at least n 4/5 triangles.

16 Previous work Theorem (Alon-Trotter) For any constant c < 1 4, there is a cn -edge graph with every edge in a triangle, but the most popular edge only in n triangles. Theorem (Edwards; Khadžiivanov-Nikiforov) Given any 1 4 n edges, there is always one in 1 6 n triangles. Theorem (Bollobás-Nikiforov) Given any 1 4 n o(n 1.4 ) edges, each of which is in a triangle, there is always some edge in at least n 4/5 triangles. Question (Erdős, 1987) Given cn edges, each of which is in a triangle, is there always some edge which is in at least n ɛ triangles, for a constant ɛ?

17 New result Theorem (Fox, L.) There are n-vertex graphs with n 4 (log (1 e n)1/6) edges, each of which is in a triangle, but with no edge in more than n 14/ log log n triangles.

18 New result Theorem (Fox, L.) There are n-vertex graphs with n 4 (log (1 e n)1/6) edges, each of which is in a triangle, but with no edge in more than n 14/ log log n triangles. Remarks: Every edge is in under n o(1) triangles. The edge density approaches 1 4 from below. Sharp transition: after edge density 1 4, some edge is in a linear number of triangles.

19 Construction materials Theorem (Hoeffding-Azuma) For any L-Lipschitz random variable X determined by n independent samples, P [ X E [X ] > t] e t L n.

20 Construction materials Theorem (Hoeffding-Azuma) For any L-Lipschitz random variable X determined by n independent samples, P [ X E [X ] > t] e t L n. Corollary: If a coin is flipped n times, the probability that the number of heads falls within n ± n is at least a constant.

21 Construction materials Theorem (Hoeffding-Azuma) For any L-Lipschitz random variable X determined by n independent samples, P [ X E [X ] > t] e t L n. Corollary: If a coin is flipped n times, the probability that the number of heads falls within n ± n is at least a constant. Classical result In even dimensions d, the Euclidean ball of radius r has ( Vol B (d) r ) = πd/ r d (d/)!.

22 Construction 0 Core tripartite graph: Take 3 copies of the lattice cube of side r in dimension d = r 5. C A B

23 Construction 0 Core tripartite graph: Take 3 copies of the lattice cube of side r in dimension d = r 5. C A B Let µ be the expected squared-distance between two random points in a single cube. A B edges correspond to squared-distances in µ ± d.

24 Construction 0 Core tripartite graph: Take 3 copies of the lattice cube of side r in dimension d = r 5. C A B Let µ be the expected squared-distance between two random points in a single cube. A B edges correspond to squared-distances in µ ± d. A C edges correspond to squared-distances in µ 4 ± d. B C edges correspond to squared-distances in µ 4 ± d.

25 Properties C A B Positive edge density: The edge density between A and B is the probability that two random points in the cube have squared-distance within µ ± d.

26 Properties C A Positive edge density: B The edge density between A and B is the probability that two random points in the cube have squared-distance within µ ± d. The squared-distance between u = (u 1,..., u d ) and v = (v 1,..., v d ) is the sum of independent (u i v i ), each ranging between 0 and r.

27 Properties C A Positive edge density: B The edge density between A and B is the probability that two random points in the cube have squared-distance within µ ± d. The squared-distance between u = (u 1,..., u d ) and v = (v 1,..., v d ) is the sum of independent (u i v i ), each ranging between 0 and r. The typical deviation from µ is r d = r 4.5 d, since d = r 5, so the A B edge density approaches 1!

28 Properties C A B Every A B edge is in a triangle: A B endpoints have squared-distance µ ± d. Their integer-rounded midpoint has squared-distance µ 4 ± d from each endpoint.

29 Properties Every A B edge is in few triangles: Given 0 = (0,..., 0) and z = (z 1,..., z d ) with z = µ ± d. Consider points x = ( z 1 + a 1,..., z d + a d )

30 Properties Every A B edge is in few triangles: Given 0 = (0,..., 0) and z = (z 1,..., z d ) with z = µ ± d. Consider points x = ( z 1 + a 1,..., z d + a d ): x = ( zi + a i z x = ( zi a i ) )

31 Properties Every A B edge is in few triangles: Given 0 = (0,..., 0) and z = (z 1,..., z d ) with z = µ ± d. Consider points x = ( z 1 + a 1,..., z d + a d ): x = ( zi + a ) i z = 4 z x = ( zi a ) i z = zi a i + a 4 1 zi a i + a 4

32 Properties Every A B edge is in few triangles: Given 0 = (0,..., 0) and z = (z 1,..., z d ) with z = µ ± d. Consider points x = ( z 1 + a 1,..., z d + a d ): x = ( zi + a ) i z = 4 z x = ( zi a ) i z = zi a i + a 4 1 zi a i + a 4 But if x and z x are both µ 4 ± d, then adding gives a 9d.

33 Properties Every A B edge is in few triangles: Given 0 = (0,..., 0) and z = (z 1,..., z d ) with z = µ ± d. Consider points x = ( z 1 + a 1,..., z d + a d ): x = ( zi + a ) i z = 4 z x = ( zi a ) i z = zi a i + a 4 1 zi a i + a 4 But if x and z x are both µ 4 ± d, then adding gives a 9d. The number of lattice points in B (d) 3 d is at most 15d

34 Properties Every A B edge is in few triangles: Given 0 = (0,..., 0) and z = (z 1,..., z d ) with z = µ ± d. Consider points x = ( z 1 + a 1,..., z d + a d ): x = ( zi + a ) i z = 4 z x = ( zi a ) i z = zi a i + a 4 1 zi a i + a 4 But if x and z x are both µ 4 ± d, then adding gives a 9d. The number of lattice points in B (d) 3 d is at most 15d r d.

35 Final construction A C B Clean up: Now every edge has few triangles; every A B edge has some.

36 Final construction A C B Clean up: Now every edge has few triangles; every A B edge has some. Delete every edge which is not part of an A B triangle. Then every edge has a triangle; total about ( n 3) edges.

37 Final construction A C B Clean up: Now every edge has few triangles; every A B edge has some. Delete every edge which is not part of an A B triangle. Then every edge has a triangle; total about ( n 3) edges. Blow up (simplification by Alon): Replace every vertex in A and B with d copies of itself. Now every edge is in at most 30 d (r) d triangles.