1. Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger

2. Introduction We have seen that in many games, Smart vs. Smart has the same outcome as Dumb vs. Dumb This talk: Smart vs. Dumb one player plays ‚against randomness‘ these are not positional games!

3. Introduction Ramsey theory: when are the edges/vertices of a graph colorable with r colors without creating a monochromatic copy of some fixed graph F ? For random graphs: solved in full generality by Ł uczak/Ruci ń ski/Voigt, 1992 (vertex colorings) Rödl/Ruci ń ski, 1995 (edge colorings)

4. Introduction ‚solved in full generality‘: Explicit threshold functions p 0 ( F, r, n ) such that In fact, p 0 ( F, r, n ) = p 0 ( F, n ), i.e., the threshold does not depend on the number of colors r [except …] The threshold behaviour is even sharper than shown here [except …] We transfer these results into an online setting, where the edges/vertices of G n, p have to be colored one by one, without seeing the entire graph.

5. The online edge-coloring game Rules: one player, called Painter start with empty graph on n vertices edges appear u.a.r. one by one and have to be colored instantly (‚online‘) either red or blue game ends when monochromatic triangle appears Question: How many edges can Painter color? Theorem (Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali, 2003): The threshold for this game is N 0 (n) = n 4/3. (easy, not main result of paper)

6. Our results Online edge-colorings: threshold for online-colorability with 2 colors for a large class of graphs F including cliques and cycles Online vertex-colorings [main focus of this talk]: threshold for online-colorability with r R 2 colors for a large class of graphs F including cliques and cycles Unlike in the offline cases, these thresholds are coarse and depend on the number of colors r.

7. The online vertex-coloring game Rules: random graph G n, p, initially hidden vertices are revealed one by one along with induced edges vertices have to be instantly (‚online‘) colored with one of r R 2 available colors. game ends when monochromatic copy of some fixed forbidden graph F appears Question: How dense can the underlying random graph be such that Painter can color all vertices a.a.s.?

8. Example F = K 3, r = 2

9. Main result (simplified) Theorem (Marciniszyn, S., 2006+) Let F be a clique or a cycle of arbitrary size. Then the threshold for the online vertex-coloring game with respect to F and with r R 2 available colors is i.e.,

10. Bounds from ‚offline‘ graph properties G n, p contains no copy of F  Painter wins with any strategy G n, p allows no r -vertex-coloring avoiding F  Painter loses with any strategy  the thresholds of these two ‚offline‘ graph properties bound p 0 ( n ) from below and above.

11. Appearance of small subgraphs Theorem (Bollobás, 1981) Let F be a non-empty graph. The threshold for the graph property ‚ G n, p contains a copy of F ‘ is where

12. Appearance of small subgraphs m ( F ) is half of the average degree of the densest subgraph of F. For ‚nice‘ graphs – e.g. for cliques or cycles – we have (such graphs are called balanced)

13. Vertex-colorings of random graphs Theorem ( Ł uczak, Ruci ń ski, Voigt, 1992) Let F be a graph and let r R 2. The threshold for the graph property ‚every r -vertex-coloring of G n, p contains a monochromatic copy of F ‘ is where

14. Vertex-colorings of random graphs For ‚nice‘ graphs – e.g. for cliques or cycles – we have (such graphs are called 1-balanced). is also the threshold for the property ‚There are more than n copies of F in G n, p ‘ Intuition: For p [ p 0, the copies of F overlap in vertices, and coloring G n, p becomes difficult.

15. For arbitrary F and r we thus have Theorem Let F be a clique or a cycle of arbitrary size. Then the threshold for the online vertex-coloring game with respect to F and with r R 1 available colors is r = 1  Small Subgraphs r    exponent tends to exponent for offline case Main result revisited

16. Lower bound ( r = 2 ) Let p ( n ) / p 0 ( F, 2, n ) be given. We need to show: There is a strategy which allows Painter to color all vertices of G n, p a.a.s.

17. Lower bound ( r = 2 ) We consider the greedy strategy: color all vertices red if feasible, blue otherwise.  after the losing move, G n, p contains a blue copy of F, every vertex of which would close a red copy of F. For F = K 4, e.g. or

18. Lower bound ( r = 2 )  Painter is safe if G n, p contains no such ‚dangerous‘ graphs. Lemma Among all dangerous graphs, F * is the one with minimal average degree, i.e., m ( F *) % m ( D ) for all dangerous graphs D. F*F* D

19. Lower bound ( r = 2 ) Corollary Let F be a clique or a cycle of arbitrary size. Playing greedily, Painter a.a.s. wins the online vertex- coloring game w.r.t. F and with two available colors if F *

20. Lower bound ( r = 3 ) Corollary Let F be a clique or a cycle of arbitrary size. Playing greedily, Painter a.a.s. wins the online vertex- coloring game w.r.t. F and with three available colors if F 3*F 3* F *

21. Lower bound Corollary Let F be a clique or a cycle of arbitrary size. Playing greedily, Painter a.a.s. wins the online vertex- coloring game w.r.t. F and with r R 2 available colors if …

22. Upper bound Let p ( n ) [ p 0 ( F, r, n ) be given. We need to show: The probability that Painter can color all vertices of G n, p tends to 0 as n  , regardless of her strategy. Proof strategy: two-round exposure & induction on r First round n / 2 vertices, Painter may see them all at once use known ‚offline‘ results Second round remaining n / 2 vertices Due to coloring of first round, for many vertices one color is excluded  induction.

23. Upper bound V1V1 V2V2 F ° 1)Painter‘s offline-coloring of V 1 creates many (w.l.o.g.) red copies of F ° 2)Depending on the edges between V 1 and V 2, these copies induce a set Base ( R ) 4 V 2 of vertices that cannot be colored red. 3)Edges between vertices of Base ( R ) are independent of 1) and 2)  Base ( R ) induces a binomial random graph Base ( R ) F  need to show: Base ( R ) is large enough for induction hypothesis to be applicable.

24. There are a.a.s. many monochromatic copies of F ‘° in V 1 provided that work (Janson, Chernoff,...)  These induce enough vertices in (w.l.o.g.) Base ( R ) such that the induction hypothesis is applicable to the binomial random graph induced by Base ( R ). Upper bound

25. The general case In general, it is smarter to greedily avoid a suitably chosen subgraph H of F instead of F itself.  general threshold function for game with r colors is where Maximization over r possibly different subgraphs H i  F, corresponding to a „smart greedy“ strategy. F H

26. A surprising example F = H 1 ] H 2 H1H1 H2H2 (lower bound only)

27. The general case Proved as a lower bound in full generality. as an upper bound assuming For any graph F, we have and

28. Back to online edge colorings Threshold is given by appearance of F *, yields threshold formula similarly to vertex case. Lower bound: Much harder to deal with overlapping outer copies! Works for arbitrary number of colors. Upper bound: Two-round exposure as in vertex case But: unclear how to setup an inductive argument to deal with r R 3 colors. F * F_F_ F °F °

29. Online edge colorings Theorem (Marciniszyn, S., Steger, 2005+) Let F be a graph that is not a tree, for which at least one F _ satisfies Then the threshold for the online edge-coloring game w.r.t. F and with two colors is F_F_

30. Online edge colorings: Trees If F is a tree, incident edges can share the same outer copy: size of minimal vertex cover seems to be crucial  greedy strategy yields different lower bound, proved as a threshold for some special cases. N 0 (n) = n 9/10

31. A not so surprising example anymore… F = K 1,4 ] P 3 K 1,4 P3P3 (lower bound only)

32. Open problems (1) More colors (edge case). Simplest open case: F = K 3, r = 3 General graphs (vertex and edge case): Replace two-round approach for upper bound by different argument (to avoid additional assumption). In particular: trees!

33. Open problems (2) Balanced Online Games Two edges appear at once, Painter has to color one red and the other one blue. Marciniszyn, Mitsche, Stojakovi ć (2005): Threshold for balanced game w.r.t. C l is Threshold for unbalanced game is Approach does not extend to cliques or trees Ongoing work: balanced online vertex colorings

34. Thank you! Questions?

36. Online vertex colorings Theorem (Marciniszyn, S., 2006+) Let F be a graph for which at least one F ° satisfies Then the threshold for the online vertex-coloring game w.r.t. F and with r R 1 colors is F°F°