Offline thresholds for online games Reto Spöhel, ETH Zürich Joint work with Michael Krivelevich and Angelika Steger TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A A AA A A

Three graph processesinvolving choices start with the empty graph on n vertices; in each step Achlioptas process: get r random edges select one of them, discard r – 1 remaining ones Ramsey process: get one random edge color it with one of r available colors Balanced Ramsey process: get r random edges color all of them, using each of r available colors exactly once Note: for r = 1 all three processes reduce to the normal random graph process without any choices involved.

Achlioptas process Achlioptas process: in each step get r random edges select one of them, discard r – 1 remaining ones Goal: Create/avoid giant component Bohman, Frieze (2001); … ; Spencer,Wormald (2007): can delay/accelerate appearance of giant by constant factors Goal: Avoid copy of F Krivelevich, Loh, Sudakov (2009): F e.g. a clique/cycle, r ¸ 2 fixed Mütze, S., Thomas (2009+): F arbitrary, r ¸ 2 fixed Goal: Create Hamilton cycle Krivelevich, Lubetzky, Sudakov (2009+): trivial lower bounds can be matched in almost all cases Torsten‘s talk

Ramsey process Ramsey process: in each step get one random edge color it with one of r available colors Goal: Avoid monochromatic copy of F Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali (2003): F = K 3, r = 2 Marciniszyn, S., Steger (2009): F e.g. a clique/cycle, r = 2 Belfrage, Mütze, S. (2009+): F e.g. a tree, r ¸ 2 fixed Goal: Create/avoid monochromatic giant component Bohman, Frieze, Krivelevich, Loh, Sudakov (2009+) Po-Shen‘s talk

Balanced Ramsey process Balanced Ramsey process: in each step get r random edges color all of them, using each of r available colors exactly once Goal: Avoid monochromatic copy of F Marciniszyn, Mitsche, Stojakovi ć (2005): F e.g. a cycle, r = 2 Prakash, S., Thomas (2009): F e.g. a cycle, r ¸ 2 fixed Goal: Create monochromatic Hamilton cycles Krivelevich, Lubetzky, Sudakov (2009+)

Three graph processesinvolving choices Achlioptas process: get r random edges select one of them, discard r – 1 remaining ones Ramsey process: get one random edge color it with one of r available colors Balanced Ramsey process: get r random edges color all of them, using each of r available colors exactly once For the rest of this talk: Goal: avoid a (monochromatic) copy of some fixed graph F r ¸ 2 is a fixed integer

Online thresholds Goal: avoid a (monochromatic) copy of some fixed graph F r ¸ 2 is a fixed integer Example: F = P 4, r = 2 : Threshold is n 9 / 10 in Ramsey process n 8 / 9 in Achlioptas process n 7 / 8 in Balanced Ramsey process … but maybe trees are special…

Online thresholds Goal: avoid a (monochromatic) copy of some fixed graph F r ¸ 2 is a fixed integer Example: F = K 4, r = 2 : Threshold is n 14 / 9 in Ramsey process n 28 / 19 in Achlioptas process and Balanced Ramsey process In general: Open whether Achlioptas and Balanced Ramsey thresholds coincide for all non-forests

Offline problems Why do the various online thresholds differ from each other? Are the differences a feature of the online setting, or are they inherited from the underlying offline problems? Offline setting corresponding to a given online process: same restrictions, but we are allowed to look at the entire random input instance at once. Ramsey problem: input instance = m random edges (sampled without replacement), i.e. random graph G n, m Rödl, Ruci ń ski (1995): For any fixed graph F and integer r ¸ 2 [except …], there exist constants c and C such that where Some well-understood exceptional cases if F is a forest

Our result Achlioptas problem/Balanced Ramsey problem: input instance = m random r-sets of edges (sampled without replacement), random r- matched graph. Krivelevich, S., Steger (2009+): For any fixed graph F and integer r ¸ 2 [except …], there exist constants c and C such that Balanced Ramsey problem: Same exceptional cases as Ramsey problem Not proved for the case where m 2 (F) is attained by a triangle Achlioptas problem: No exceptional or unproven cases! fully understood, both online and offline

Achlioptas: the full picture F = offline online n1n1 r=1r=1 n1n1 r=1r=1 Erd ő s, Rényi (1960) Bollobás (1981) n 1.5 r ¸ 2 Krivelevich, S., Steger (2009+) n … r=3r=3 n … r=4r=4 n 1.2 r=2r=2 n … r= 1000 Krivelevich, Loh, Sudakov (2009) Mütze, S., Thomas (2009+) Torsten‘s talk

Our result: conclusions For ‚most‘ graphs F (e.g. K l, C l, P l ; l ¸ 4 ) and any r ¸ 2, the offline thresholds of the Achlioptas problem, the Ramsey problem, and the Balanced Ramsey problem coincide at (in order of magnitude). In particular, the order of magnitude of the offline thresholds does not depend on r, in contrast to what happens in online settings Conclusions: The differences in the online thresholds are not inherited from the underlying offline problems, they stem from the online setting. The online problems are much more susceptible to slight variations of the rules than the offline problem!

About the proofs Note: Balanced Ramsey is harder than Achlioptas Suffices to show Upper bound for Achlioptas problem Lower bound for Balanced Ramsey problem

Lower Bound Proof Inspired by Rödl/Ruci ń ski LB proof for Ramsey problem Key insight in their proof: Consider the hypergraph on vertex set E ( G n, m ) with hyperedges given by the copies of F For, this hypergraph a.a.s. has unicyclic components of at most logarithmic size (w.l.o.g. F strictly 2- balanced) Key insight in our proof: The same is true for the analogously defined hypergraph which has r -sets of edges as its vertex set.

Upper Bound Proof We need to prove: For, a.a.s. every Achlioptas subgraph of contains a copy of F.

Upper Bound Proof In fact we prove: For, a.a.s. every Achlioptas subgraph of contains ‚many‘ copies of F. ‚many‘: a constant fraction of the expected number in G n,m This can be shown by induction on e F, using a two-round approach in each induction step similar to (but easier than) Rödl/Ruci ń ski upper bound proof

Summary & open questions For ‚most‘ graphs F (e.g. K l, C l, P l ; l ¸ 4 ) and any r ¸ 2, the offline thresholds of the Achlioptas problem, the Ramsey problem, and the Balanced Ramsey problem coincide at (in order of magnitude). Open questions: Many open questions for Ramsey and Balanced Ramsey online e.g. online Ramsey threshold for F = K 3, r = 3 unknown What happens if r = r(n) is a (slowly) growing function? Opposite problem: creating a copy of F as quickly as possible some preliminary results for Achlioptas process (joint work with M. Krivelevich)

Offline thresholds for other online games Avoiding a giant component (r = 2 ): Bohman, Kim (2006): Achlioptas offline threshold is c 1 n, where c 1 ¼ S., Steger, Thomas (2009+): Ramsey offline threshold= Balanced Ramsey offline threshold is c 2 n, where c 2 ¼ Henning‘s talk