Packing nearly optimal Ramsey R(3, t) graphs

Transcription

1 Packing nearly optimal Ramsey R(3, t) graphs He Guo Joint work with Lutz Warnke

2 Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue edge-coloring of complete n-vertex graph K n contains red K s or blue K t Major problem in combinatorics: determining asymptotics Testbed for new proof techniques/methods: Alteration, LLL, Concentration Ineq., Semi-Random, Differential Eq. Celebrated Result (Ajtai-Komlós-Szemerédi Kim 1995) R(3, t) = Θ(t 2 / log t) Lower bound harder: Kim received Fulkerson Prize 1997 R(3, t) = Ω(t 2 /(log t) 2 ) already by Erdős in 1961 Topic of this talk Extension of Kim-result (implies asymptotics of other Ramsey parameter)

3 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a

4 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a Why this result is difficult? Standard approach (alteration): in G n,p, try to remove one edge of all s Facts: w.h.p. #edges = Θ(n 2 p) #K 3 s = Θ(n 3 p 3 ) =Θ(n 2 p np 2 ) #edges p = ε/ n Max ISET of G n,p 2 log n p! = C n log n n log n

5 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a -free process: add one random edge in each step closed (can not add) open (can add)

6 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a -free process: add one random edge in each step closed (can not add) open (can add)

7 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a -free process: add one random edge in each step closed (can add) open (can add)

8 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a -free process: add one random edge in each step closed (can not add) open (can add)

9 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a -free process: add one random edge in each step closed (can not add) open (can add)

10 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a -free process: add one random edge in each step closed (can not add) open (can add)

11 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a Semi-random variation: add many random-like edges in each step closed (can not add) open (can add)

12 Main Result: nearly optimal R(3, t) graphs Kim (1995) + Bohman (2008): one nearly optimal R(3, t) graph Both find an n-vertex graph G K n such that G is -free with independence number α(g) C n log n Using (semi-random variation of) -free process: greedily add random edges that do not close a G., Warnke (2017+): almost packing of nearly optimal R(3, t) graphs Given ε > 0, we find edge-disjoint graphs (G i ) i I with G i K n such that (a) each G i is -free with α(g i ) C ε n log n (b) the union of the G i contains (1 ε) ( n 2) edges Using simple polynomial-time randomized algorithm: sequentially choose G i via semi-random variation of -free process

13 Main Result: nearly optimal R(3, t) graphs G., Warnke (2017+): almost packing of nearly optimal R(3, t) graphs Given ε > 0, we find edge-disjoint graphs (G i ) i I with G i K n such that (a) each G i is -free with α(g i ) C ε n log n (b) the union of the G i contains (1 ε) ( n 2) edges Using simple polynomial-time randomized algorithm: sequentially choose G i via semi-random variation of -free process Motivation: why should we care? Natural packing extension of Kim s result Technical challenge: controlling errors over Θ( n/ log n) iterations Establishes Ramsey-Theory conjecture by Fox et.al. (cf. next slides)

14 Ramsey Theory with r 2 colors G (H) r : any r-coloring of E(G) has monochromatic copy of H Ramsey theory studying properties of r-ramsey minimal graphs M r (H) := all graphs G that are r-ramsey minimal for H (i.e., G (H) r and G (H) r for all G G) min G Mr (K k ) v(g) = Ramsey number min G Mr (K k ) e(g) = Size Ramsey number Minimum degree of r-ramsey minimal graphs (Burr, Erdős, Lovász 1976) s r (H) := min G Mr (H) δ(g) s 2 (K k ) = (k 1) 2 : Burr, Erdős, Lovász (1976) s 2 (H) = 2δ(H) 1: for many bipartite H (trees, K a,b, etc) Fox, Lin (2006) + Szabó, Zumstein, Zürcher (2010) s r (K k ) = Θ k (r 2 ): Fox, Grinshpun, Liebenau, Person, Szabó (2015)

15 Ramsey Conjecture of Fox et.al. Minimum degree of minimal r-ramsey graphs (Burr, Erdős, Lovász 1976) s r (K k ) := min G Mr (K k )δ(g) cr 2 log r s r (K 3 ) Cr 2 (log r) 2 by FGLPS (2015) Conjecture (Fox, Grinshpun, Liebenau, Person, Szabo, 2015) s r (K 3 ) = O(r 2 log r) They suggested to pack G i sequentially via -free process (their weaker upper bound relies on sequential LLL argument) Conj. True (G., Warnke, 2017+): corollary of our main packing result Implies s r (K 3 ) = Θ(r 2 log r) For technical reasons: use semi-random variation of -free process

16 Ramsey Conjecture of Fox et.al. Minimum degree of minimal r-ramsey graphs (Burr, Erdős, Lovász 1976) s r (K k ) := min G Mr (K k )δ(g) cr 2 log r s r (K 3 ) Cr 2 (log r) 2 by FGLPS (2015) Conjecture (Fox, Grinshpun, Liebenau, Person, Szabo, 2015) s r (K 3 ) = O(r 2 log r) They suggested to pack G i sequentially via -free process (their weaker upper bound relies on sequential LLL argument) Conj. True (G., Warnke, 2017+): corollary of our main packing result Implies s r (K 3 ) = Θ(r 2 log r) For technical reasons: use semi-random variation of -free process

17 Glimpse of the proof Main-Technical-Result: find random-like -free subgraph G H Let ϱ := β(log n)/n and s := C ε n log n. If H Kn is such that e H (A, B) ε A B for all disjoint sets A, B of size s, then we can find -free G H with e G (A, B) = (1 ± δ) ϱ e H (A, B) for all disjoint A, B of size s. Proof based on semi-random variation of -free process: Do not require degree/codegree regularity of H Self-stabilization mechanism built into process (to control errors) Tools: Bounded-Differences-Ineq. and Upper-Tail-Ineq. of Warnke

18 Glimpse of the proof Main-Technical-Result: find random-like -free subgraph G H Let ϱ := β(log n)/n and s := C ε n log n. If H Kn is such that e H (A, B) ε A B for all disjoint sets A, B of size s, then we can find -free G H with e G (A, B) = (1 ± δ) ϱ e H (A, B) for all disjoint A, B of size s. Implies packing result: (maintaining e Hi (A, B) bounds inductively) Start with H 0 = K n Sequentially choose G i H i and set H i+1 = H i \ G i Stop when e HI (A, B) ε A B holds

19 Semi-random construction of -free subgraph To construct triangle-free T J, we iteratively keep track of E j : random set of edges T j E j : -free and T j E j O j {all e E j that don t form a with any two edges of E j } Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free

20 Semi-random construction of -free subgraph To construct triangle-free T J, we iteratively keep track of E j : random set of edges T j E j : -free and T j E j O j {all e E j that don t form a with any two edges of E j } Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free

21 Random edge-set Γ i+1 and edge-set E j+1 Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free Definition of Γ j+1 and E j+1 Γ j+1 O j : p-random subset of O j E j+1 = E j Γ j+1 Why can we ensure Γ j+1 Γ j+1? Γ j+1 small very few new s created in E j Γ j+1 hence removal of few edges destroys all new s

22 Random edge-set Γ i+1 and edge-set E j+1 Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free Definition of Γ j+1 and E j+1 Γ j+1 O j : p-random subset of O j E j+1 = E j Γ j+1 Why can we ensure Γ j+1 Γ j+1? Γ j+1 small very few new s created in E j Γ j+1 hence removal of few edges destroys all new s

23 Random edge-set Γ i+1 and edge-set E j+1 Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free Definition of Γ j+1 and E j+1 Γ j+1 O j : p-random subset of O j E j+1 = E j Γ j+1 Why can we ensure Γ j+1 Γ j+1? Γ j+1 small very few new s created in E j Γ j+1 hence removal of few edges destroys all new s

24 Finding -free Γ j+1 Γ j+1 Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free E j Γ j+1 can create new s: Bad pairs Bad triples Alteration to destroy new s: Γ j+1 = Γ j+1 \ D j+1 D j+1 = edges of a maximal edge-disjoint collection of bad pairs/triples easier to analyze than removing 1 edge from each new T j+1 = T j Γ j+1 is -free by maximality of D j+1

25 Finding -free Γ j+1 Γ j+1 Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free E j Γ j+1 can create new s: Bad pairs Bad triples Alteration to destroy new s: Γ j+1 = Γ j+1 \ D j+1 D j+1 = edges of a maximal edge-disjoint collection of bad pairs/triples easier to analyze than removing 1 edge from each new T j+1 = T j Γ j+1 is -free by maximality of D j+1

26 Open edges: effect of closed edges Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free Updating open edges that can still be added O j+1 = O j \(Γ j+1 { closed edges } {extra edges for technical reasons}). Closed edge forms a triangle with two edges in E j+1 = E j Γ j+1. Closed edge Closed edge

27 Open edges: self-stabilization mechanism Updating open edges that can still be added O j+1 = O j \ (Γ j+1 { closed edges } {extra random edges}).

28 Open edges: self-stabilization mechanism Updating open edges that can still be added O j+1 = O j \ (Γ j+1 { closed edges } {extra random edges}). Y e (j) = # edges whose addition to E j+1 will close e u adding any of green edges closes e = {u, v} v Self-stabilization: make P(closed) equal for all e (independent of history) P(e not closed in next step of iteration) (1 p) Ye(j) P(e not (closed or extra edge)) (1 p) Ye(j) (1 q e )! = same for all e

29 Semi-random construction of -free subgraph To construct triangle-free T J, we iteratively keep track of E j : random set of edges T j E j : -free and T j E j O j {all e E j that don t form a with any two edges of E j } Idea of each step (1) Generate few random edges Γ j+1 O j (2) Alteration: find Γ j+1 Γ j+1 s.t. T j+1 = T j Γ j+1 (3) Update O j+1 O j \ Γ j+1 remains -free

30 Number of edges between two large sets Assume we can show O j (A, B) q j A B, where q j = Ψ (jσ), for O 0 = H = K n. Use p = σ/ n, then we can approximate T J (A, B)

31 Number of edges between two large sets Assume we can show O j (A, B) q j A B, where q j = Ψ (jσ), for O 0 = H = K n. Use p = σ/ n, then we can approximate T J (A, B) T J (A, B) = T j+1 (A, B) \ T j Γ j+1 (A, B) 0 j<j 0 j<j 1 n Jσ 0 j<j p O j (A, B) 1 n 0 0 j<j σq j A B Ψ (x)dx A B Ψ(Jσ) A B n β(log n) A B = ϱ A B n

32 A technical difficulty Difficulty of tracking O j (A, B) Choosing one edge into Γ j+1 may cause large change of O j (A, B) : N Ej (v 1 ) A v 1 v 2 B

33 A technical difficulty Difficulty of tracking O j (A, B) Choosing one edge into Γ j+1 may cause large change of O j (A, B) : N Ej (v 1 ) A v 1 v 2 B

34 Summary G., Warnke (2017+): almost packing of nearly optimal R(3, t) graphs Given ε > 0, we find edge-disjoint graphs (G i ) i I with G i K n such that (a) each G i is -free with α(g i ) C ε n log n (b) the union of the G i contains (1 ε) ( n 2) edges Remarks Natural algorithmic packing version of Kim s R(3, t) construction Establishes s r (K 3 ) = Θ(r 2 log r) asymptotics conjectured by Fox et.al. Questions Further applications of the K 3 -free packing result? Generalization of packing-result to K k -free graphs worth effort?