Ramsey games in random graphs PhD defense 22 nd Nov 2011 Torsten Mütze TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

Playing games Time Board games Egypt, 3000 BC Ball games Mesoamerica, 1000 BC Card games China, 9th century Europe, 14th century Computer games 20th century

Combinatorial games Combinatorial games: perfect and complete information Outcome can in principle be predicted by exhaustively enumerating all possible ways the game may evolve Positional games: Players alternately claim elements of some board Probabilistic methods very powerful in dealing with ‘combinatorial chaos’ in positional games The probabilistic intuition in positional games [Chvátal, Erdős ‘78], [Beck ‘93, …] : Chess NimTic-Tac-Toe Hex game tree ‘Combinatorial chaos’ huge!!! clever vs. clever = random vs. random

Main contribution of this thesis Main contribution of this thesis: Build a similar bridge between two previously disconnected worlds Positional games where the goal is to avoid some given local substructure Benefit: Transfer insights and techniques between the two worlds and derive new results in each of them clever vs. random Probabilisti c one-player games Deterministic two-player games clever vs. clever The probabilistic intuition in positional games [Chvátal, Erdős ‘78], [Beck ‘93, …] : clever vs. clever = random vs. random

Probabilistic one-player avoidance games (clever vs. random) Edge-coloring game [Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali ‘03] The probabilistic world Painter vs. random graph process n Goal: Avoid monochromatic copies of F for as long as possible F = K 3

Probabilistic one-player avoidance games (clever vs. random) Edge-coloring game [Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali ‘03] Vertex-coloring game [Marciniszyn, Spöhel ‘10] The probabilistic world Painter vs. random graph process Goal: Avoid monochromatic copies of F for as long as possible n F = K 3

Probabilistic one-player avoidance games (clever vs. random) Edge-coloring game [Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali ‘03] Vertex-coloring game [Marciniszyn, Spöhel ‘10] Achlioptas game [Krivelevich, Loh, Sudakov ‘09] The probabilistic world Chooser vs. random graph process Goal: Avoid the appearance of F for as long as possible n Painter vs. random graph process Goal: Avoid monochromatic copies of F for as long as possible F = K

The deterministic world Deterministic two-player avoidance games (clever vs. clever) Ramsey-game [Beck ‘83] [Kurek, Ruci ń ski ‘05] Impose restrictions on Builder [Grytczuk, Haluszczak, Kierstead ‘04] : Restrict to graphs with chromatic number at most  Builder can still enforce a monochromatic copy of Restrict to forests  Builder can still enforce a monochromatic copy of any forest Painter vs. Builder Goals: Avoid / enforce monochromatic copies of F for as long / as quickly as possible ? Yes for infinitely many values of [Conlon ‘10] Online size-Ramsey number := minimum number of steps necessary for Builder to win

A bridge between the two worlds Idea: Replace the random graph process by an adversary with a suitable density restriction Observation: Winning strategies for adversary in the deterministic game yield upper bounds for the duration of the probabilistic game Corresponding lower bound statement much harder to prove Density restriction: Builder must adhere to for all subgraphs H Concrete results for these games later! Painter vs. random graph processBuilder Edge-coloring game Vertex-coloring game H H

Achlioptas game Complete solution, i.e., threshold functions for arbitrary fixed F, presented in [M., Spöhel, Thomas ‘10], disproving a conjecture from [Krivelevich, Loh, Sudakov ‘09] Implicit in the analysis: Chooser vs. random graph process Presenter A bridge between the two worlds Idea: Replace the random graph process by an adversary with a suitable density restriction Observation: Winning strategies for adversary in the deterministic game yield upper bounds for the duration of the probabilistic game Corresponding lower bound statement much harder to prove Density restriction: Presenter must adhere to for all subgraphs H

A bridge between the two worlds Idea: Replace the random graph process by an adversary with a suitable density restriction Observation: Winning strategies for adversary in the deterministic game yield upper bounds for the duration of the probabilistic game Corresponding lower bound statement much harder to prove This upper bound technique extends straightforwardly to similar avoidance games played on random hypergraphs or random subsets of integers

The edge-coloring game [Friedgut, Kohayakawa, Rödl, Ruciński, Tetali ‘03] : The threshold (typical duration) of the game with F = K 3 and r =2 colors is [Marciniszyn, Spöhel, Steger ‘05] : Explicit threshold functions for F (e.g.) a clique or a cycle and r =2 colors n For any : there is a Painter strategy that succeeds whp. For any : every Painter strategy fails whp. N = number of steps Threshold Painter vs. random graph process

The edge-coloring game [Belfrage, M., Spöhel ‘11+] : New upper bound approach… Successfully applied by [Balogh, Butterfield ‘10] to derive the first nontrivial upper bounds for F = K 3 and r R 3 colors Density restriction: Builder must adhere to for all subgraphs H Painter vs. random graph process n Builder H Theorem: If Builder can enforce a monochromatic copy of F in the deterministic game with r colors and density restriction d, then the threshold of the probabilistic game is bounded by

Proof idea: Well-known: If F is a fixed graph with for all, then for any, whp. after N steps the evolving random graph contains many copies of F. Can be adapted to: If T is a fixed Builder strategy respecting a density restriction of d, then for any, whp. after N steps the evolving random graph behaves exactly like T in many places on the board. The edge-coloring game Theorem: If Builder can enforce a monochromatic copy of F in the deterministic game with r colors and density restriction d, then the threshold of the probabilistic game is bounded by

The edge-coloring game Theorem: If Builder can enforce a monochromatic copy of F in the deterministic game with r colors and density restriction d, then the threshold of the probabilistic game is bounded by Upper bound technique translates straightforwardly to other settings (vertex-coloring game, Achlioptas game, random hypergraphs, random subsets of integers etc.) Corresponding lower bound statements require problem-specific work (if provable at all) Open problem: Define the online Ramsey density as Is it true that ?

The vertex-coloring game Painter vs. random graph p = edge probability Threshold For any : there is a Painter strategy that succeeds whp. For any : every Painter strategy fails whp. [Marciniszyn, Spöhel ‘10] : Explicit threshold functions for F (e.g.) a clique or a cycle and r R 2 colors ? For these graphs, a simple greedy strategy is optimal The greedy strategy is not optimal for every graph, the general case remained open

The vertex-coloring game [M., Rast, Spöhel ‘11+] : For any fixed F and r, we can compute a rational number such that the threshold is Painter vs. random graph ! For these graphs, a simple greedy strategy is optimal We solve the problem in full generality Builder Density restriction: Builder must adhere to for all subgraphs H H

The vertex-coloring game Painter vs. random graph Builder H d Builder can enforce F monochromatically in finitely many steps Painter can avoid monochromatic copies of F indefinitely Define the online vertex-Ramsey density as Density restriction: Builder must adhere to for all subgraphs H

Painter vs. Builder Painter vs. random graph Theorem 1: For any F and r is computable is rational infimum attained as minimum Theorem 2: For any fixed F and r, the threshold of the probabilistic one-player game is focus for next few slides

Painter vs. Builder – Remarks Theorem 1: For any F and r is computable is rational infimum attained as minimum …nor for the two edge-coloring analogues [Rödl, Ruci ń ski ‘93], [Kurek, Ruci ń ski ‘05], [Belfrage, M., Spöhel ‘11+] zloty prize money for [Rödl, Ruci ń ski ‘93] None of those three statements is known for the offline quantity [Kurek, Ruci ń ski ‘94]

Painter vs. Builder – Remarks The running time of our procedure for computing is doubly exponential in v ( F )… With the help of a computer we determined exactly for all graphs F on up to 9 vertices for F a path on up to 45 vertices Theorem 1: For any F and r is computable is rational infimum attained as minimum

Painter vs. Builder Painter vs. random graph Theorem 1: For any F and r is computable is rational infimum attained as minimum Theorem 2: For any fixed F and r, the threshold of the probabilistic one-player game is focus for next few slides

Painter vs. random graph – Remarks In the asymptotic setting of Theorem 2, computing is a constant-size computation! So is computing the optimal Painter and Builder strategies for the deterministic game For some of Painter’s optimal strategies in the deterministic game, we can show that they also work in the probabilistic game  polynomial-time coloring algorithms that succeed whp. in coloring G n, p online for any Theorem 2: For any fixed F and r, the threshold of the probabilistic one-player game is

Painter vs. random graph – Proof ideas Upper bound: Use our general approach to translate an optimal Builder strategy from the deterministic game to an upper bound of for the probabilistic game Theorem 2: For any fixed F and r, the threshold of the probabilistic one-player game is

Painter vs. random graph – Proof ideas Theorem 2: For any fixed F and r, the threshold of the probabilistic one-player game is Lower bound : Much more involved… Playing ‘just as in the deterministic game’ does not necessarily work for Painter! Reason: the probabilistic process with p ¿ n -1/ d respects a density restriction of d only locally (the entire random graph has an expected density of £ ( np ) ) To overcome this issue, we need to really understand the deterministic game and the structure of Painter’s and Builder’s optimal strategies.

The path-avoidance vertex-coloring game So is computable, but what is its value for natural families of graphs F like,,,,,, ? [M., Spöhel ’11] : exhibits a surprisingly complicated behavior Greedy strategy fails quite badly Evidence that a general closed formula for does not exist Simple closed formulas ( [Marciniszyn, Spöhel ‘10] ) Reason: Greedy strategy is optimal

The path-avoidance vertex-coloring game Forests F : greedy lower bound Theore m: Asymptotics for large ? vertices smallest k s.t. Builder can enforce F while not building cycles and only trees with at most k vertices

Thank you!

deterministic probabilistic online offline edges vertices deterministic probabilistic online offline edges vertices [Rödl, Ruciński ‘93, ‘95] edges vertices [ Ł uczak, Ruciński, Voigt ‘92] Vertex-Ramsey density Ramsey density [RR‘93, KR‘94] [RR‘93, KR‘05] ORD OVRD ? ? ? ? 3-D-Games

Random subsets of integers Numbers in {1,…, n } are revealed in uniform random order Player Painter has to color each of them red or blue Painter’s goal: Avoid monochromatic arithmetic progression of length 3, say Question: What is the typical number of steps for which Painter can survive? Upper bound technique: 1 2 n Painter vs. random permutation Presenter Density restriction: Presenter must adhere to for all subsequences  xy z 3 2 =1.5 Example:

Issues in the edge-coloring game We need to really understand the deterministic game and the structure of Painter’s and Builder’s optimal strategies Vertex-coloring game Edge-coloring game Builder only needs to focus on monochromatic subgraphs of F => systematic exploration of Builder’s options ? Achlioptas game