1. Online Graph Avoidance Games in Random Graphs Reto Spöhel Diploma Thesis Supervisors: Martin Marciniszyn, Angelika Steger

2. Outline The online graph avoidance game Rules and known result Main result Proof Lower bound Upper bound Outlook The game with more colors

3. The online graph avoidance game Rules: one player, called the Painter starts with the empty graph on n vertices edges appear one by one u.a.r. and have to be instantly (‚online‘) colored either red or blue The game ends as soon as the Painter closes a monochromatic copy of a fixed forbidden graph F. Question: How many edges can the Painter typically color?

4. Example n = 6, F = K 3 Length of game: N = 9 edges

5. Known result Theorem (Friedgut et al., 2003) The threshold for the online triangle avoidance game with two colors is i.e.,

6. Main result Theorem (Main result for cliques) Let F be a clique of arbitrary size. Then the threshold for the online F -avoidance game with two colors is

7. Bounds from ‚offline‘ graph properties Let G ( n, N ) denote the graph on n vertices obtained by inserting N edges one by one, u.a.r. G ( n, N ) is distributed uniformly over all graphs with n vertices and N edges. Game ends at time N : G ( n, N ) contains a copy of F. G ( n, N - 1 ) can be 2-coloured avoiding monochromatic copies of F. ) the thresholds of these two ‚offline‘ graph properties bound N 0 ( n ) from below and above.

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

9. Appearance of small subgraphs m ( F ) is the edge density of the densest subgraph of F. For ‚nice‘ graphs – e.g. for cliques – we have (such graphs are called balanced)

10. Ramsey theory in random graphs Theorem (Rödl/Rucinski, 1995) Let F be a graph which is not a star forest. The threshold for the graph property ‚every 2-coloring of G ( n, N ) contains a monochromatic copy of F ‘ is where

11. Ramsey theory in random graphs For ‚nice‘ graphs – e.g. for cliques – we have (such graphs are called 2-balanced). is also the threshold for the graph property ‚There are more copies of F than edges in G ( n, N )‘ Intuition: For N À N 0 this forces the copies of F to overlap substantially, and coloring G ( n, N ) becomes difficult.

12. Main result revisited For arbitrary F we thus have Theorem (Main result for cliques) Let F be a clique of arbitrary size. Then the threshold for the online F -avoidance game with two colors is

13. Lower bound Let N ( n ) ¿ N 0 ( n ) be arbitrary. We need to show: There is a strategy which allows the Painter to color N ( n ) edges with probability tending to 1 as n !1. We consider the greedy strategy: color all edges red if feasible, blue otherwise. Proof strategy: Reduce the event that the Painter fails to the appearance of a certain dangerous graph F * in G ( n, N ). Apply ‘small subgraphs’ theorem. [‘asymptotically almost surely, a.a.s.’]

14. Lower bound Analysis of the greedy strategy: color all edges red if feasible, blue otherwise. ) after the losing move, the graph contains a blue copy of F, every edge of which would close a red copy of F. For F = K 4, e.g. or

15. Lower bound ) A greedy Painter loses if the edges of one of these dangerous graphs appear in a bad order. ) The Painter is secure as long as none of these graphs appear in G( n, N ).

16. Lower bound Lemma F * is a.a.s. the first dangerous graph which appears in G ( n, N ). Proof idea: By the ‚small subgraphs‘ theorem, we need to show m ( F *) < m ( D ) for all other dangerous graphs D. D F *

17. Lower bound Construct a given dangerous graph D inductively by merging edges and vertices of F *… D …and use amortized analysis to prove that m ( D ) > m ( F *). F * D

18. Lower bound Corollary (Lower bound) Let F be a clique of arbitrary size. Playing greedily, the Painter can a.a.s. color any edges. F *

19. Upper bound Let N ( n ) À N 0 ( n ) be arbitrary. We need to show: For every strategy of the Painter, the probability that she can color N ( n ) edges tends to 0 as n !1.

20. Upper bound Proof strategy: two-round exposure First round N 0 edges, Painter may see them all at once a.a.s. every coloring creates many ‘threats‘. use results from (offline) Ramsey theory Second round remaining N 1 À N 0 edges the Painter will a.a.s. encounter a threat and hence lose the game use second moment method

21. Consider G ( n, N 0 ) = R [ B, the 2-coloring assigned to the first N 0 edges by the Painter. Base( R ) := {edges which would close a red copy of F } All edges in Base( R ) have to be colored blue if presented to the Painter. ) Painter loses in second round as soon as she is given a copy of F ½ Base( R ). Second round: the base graph

22. Second round: threats Threats = copies of F in Base( R ) or Base( B ) If there are many [=: M ] threats after the first round, by the second moment method a.a.s one of them is hit in the second round. many: enough to ensure that  [ X ] ! 1, where X := number of threats hit in second round. second moment method: Var[ X ] ! 0 fast enough to guarantee that X is a.a.s. close to  [ X ].

23. Threats = copies of F ½ Base( R ) are induced by copies of ½ R. ) We want to find many copies of in either R or B. First round: looking for threats

24. A counting version Theorem (Rödl/Rucinski, 1995) Let H be any non-empty graph, and let Then there is a constant c = c ( H ) > 0 such that a.a.s every 2-coloring of G ( n, N ) contains at least monochromatic copies of H. We will apply this with H = and N=N 0 to find many monochromatic copies of in G ( n, N 0 ).

25. ) There are a.a.s. M monochromatic copies of in G ( n, N 0 ), provided that These induce M threats ) a.a.s. the Painter loses in the second round. First round: finding the threats

26. A simplification Lemma where := ‘ F with one edge removed‘.

27. Upper bound Corollary (upper bound) Let F be a clique of arbitrary size. Regardless of her strategy, the Painter is a.a.s not able to colour any edges.

28. Main result Theorem (Main result) Let F be a 2-balanced and regular graph for which at least one satisfies Then the threshold for the online F -avoidance game with two colors is

29. Special Cases Corollary (Clique avoidance games) For l ¸ 2, the threshold for the online K l -avoidance game with two colors is Corollary (Cycle avoidance games) For l ¸ 3, the threshold for the online C l -avoidance game with two colors is

30. Outlook: the game with more colors Same rules, but Painter now has r ¸ 2 colors available. There is still an obvious greedy strategy: number the colors from 1 to r always use the lowest number color which does not close a monochromatic copy of F. F e.g. a clique ) there is a unique ‚basic dangerous graph‘

31. F = K 3 ; r = 3 ; colors: yellow, red, blue Example: the graph

32. Outlook: the game with more colors If is the first dangerous graph which appears in G ( n, N ), the greedy strategy ensures a.a.s. survival up to any edges.

33. Outlook: the game with more colors We believe: For l ¸ 2 and r ¸ 1, the threshold for the online K l -avoidance game with r colors is