Ramsey games inrandom graphs

PhD defense22nd Nov 2011Torsten Mütze

Playing games

Time

Board games

Egypt, 3000 BC

Ball games

Mesoamerica, 1000 BC

Card games

China, 9th centuryEurope, 14th century

Computer games

20th century

Combinatorial games• Combinatorial games: perfect and complete information

•Outcome can in principle be predicted byexhaustively enumerating all possibleways 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

Nim Tic-Tac-Toe

Hex

gametree

‘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 toavoid some given local substructure

•Benefit: Transfer insights and techniques between the two worlds and derive new results in each of them

clever vs. random

Probabilisticone-playergames

Deterministictwo-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, Ruciński, Tetali ‘03]

The probabilistic world

Painter vs.random graph processn Goal: Avoid monochromatic

copies of F for as long aspossible

F = K3

• Probabilistic one-player avoidance games (clever vs. random)

• Edge-coloring game[Friedgut, Kohayakawa, Rödl, Ruciński, Tetali ‘03]

• Vertex-coloring game[Marciniszyn, Spöhel ‘10]

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1

3

4

5

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8

The probabilistic world

Painter vs.random graph processGoal: Avoid monochromatic

copies of F for as long aspossible

n

F = K3

• Probabilistic one-player avoidance games (clever vs. random)

• Edge-coloring game[Friedgut, Kohayakawa, Rödl, Ruciński, 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 appearanceof F for as long as possible

n

Painter vs.random graph processGoal: Avoid monochromatic

copies of F for as long aspossible

F = K3

2

1

3

4

5

6

7

8

The deterministic world• Deterministic two-player avoidance games (clever vs. clever)

• Ramsey-game[Beck ‘83][Kurek, Ruciński ‘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. BuilderGoals: Avoid / enforce

monochromatic copies of Ffor as long / as quickly as possible

? Yes for infinitely many values of [Conlon ‘10]

Online size-Ramsey number := minimum number of stepsnecessary 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 resultsfor these games later!

Painter vs. random graph process

Builder

Edge-coloring gameVertex-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 thegame with F = K3 and r=2 colorsis

• [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 stepsThreshold

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 = K3 and rR3 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 inthe 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 evolvingrandom graph contains many copies of F.

•Can be adapted to:If T is a fixed Builder strategy respecting a density restrictionof 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 inthe 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 inthe 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 othersettings (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 gamePainter vs.random graph

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p = edge probabilityThreshold

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 cliqueor a cycle and rR2 colors

?• For these graphs, a simple

greedy strategy is optimal

• The greedy strategy is not optimalfor every graph, the general case remained open

The vertex-coloring game• [M., Rast, Spöhel ‘11+]:

For any fixed F and r, we cancompute a rational numbersuch 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 gameThe vertex-coloring gamePainter vs.random graphBuilder

H

d

Builder can enforceF monochromaticallyin finitely many steps

Painter can avoidmonochromaticcopies of Findefinitely

• 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

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, Ruciński ‘93], [Kurek, Ruciński ‘05], [Belfrage, M., Spöhel ‘11+]

• 400.000 zloty prize money for

[Rödl, Ruciński ‘93]

• None of those three statements is known for the offline quantity

[Kurek, Ruciński ‘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 nextfew slides

focus for nextfew 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 Gn, 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 whatis its value for natural families of graphsF like , , , , , , ?

• [M., Spöhel ’11]:

• exhibits a surprisingly complicated behavior

•Greedy strategy fails quite badly

•Evidence that a general closed formula fordoes not exist

• Simple closed formulas([Marciniszyn, Spöhel ‘10])

• Reason: Greedy strategyis optimal

The path-avoidance vertex-coloring game

• Forests F :

greedy lower bound

Theorem:

• Asymptotics for large ?

vertices•

smallest k s.t. Builder can enforce F while not buildingcycles and only trees with at most k vertices

Thank you!