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

2. Introduction Ramsey theory: when are the vertices/edges 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 Luczak/Rucinski/Voigt, 1992 (vertex colorings) Rödl/Rucinski, 1995 (edge colorings)

3. Introduction ‚solved in full generality‘: There is a threshold function p 0 ( F, r, n ) such that In fact, p 0 ( F, r, n ) = p 0 ( F, n ) in almost all cases, i.e., the threshold does not depend on the number of colors r (in order of magnitude). We transfer these results into an online setting, where vertices/edges have to be colored one by one, without seeing the entire graph.

4. Overview of results Online vertex-colorings (this talk): threshold for online-colorability with r R 2 colors for a large class of graphs F including cliques and cycles Online edge-colorings: threshold for online-colorability with 2 colors for a large class of graphs F including cliques and cycles only non-trivial lower bounds for more than 2 colors. Unlike in the offline case, these thresholds depend on the number of colors r.

5. The online vertex-coloring game Rules: one player, called Painter 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 as soon as Painter closes a monochromatic copy of some fixed forbidden graph F. Question: How dense can the underlying random graph be such that Painter can color all vertices a.a.s.?

6. Example F = K 3, r = 2

7. Main result 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.,

8. 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.

9. 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

10. 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)

11. Vertex-colorings of random graphs Theorem (Luczak/Rucinski/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

12. 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.

13. 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

14. 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. We consider the greedy strategy: color all vertices red if feasible, blue otherwise. Proof strategy: reduce the event that Painter fails to the appearance of a certain dangerous graph F * in G n, p. apply Small Subgraphs Theorem.

15. Lower bound ( r = 2 ) Analysis of 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

16. 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

17. 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 *

18. 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 *

19. 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 …

20. 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 results from (offline) Ramsey theory Second round remaining n / 2 vertices Further restrictions due to coloring of first round For many vertices one color excluded  induction.

21. Upper bound V1V1 V2V2 F°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.

22. 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

23. Main result revisited Theorem Let F be a 1-balanced 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° F *

24. Outlook 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 Proved as a lower bound in full generality Proved as an upper bound assuming

25. Thank you! Questions?

26. Online edge colorings Theorem (Marciniszyn/S./Steger, 2005) Let F be a 2-balanced 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_F_