Bounds on Ramsey Games via Alterations

09/06/2019 ∙ by He Guo, et al. ∙ Georgia Institute of Technology 0

This note contains a refined alteration approach for constructing H-free graphs: we show that removing all edges in H-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdos and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers.

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1 Introduction

The probabilistic method is a widely-used tool in discrete mathematics. Many of its powerful approaches have been developed in the pursuit of understanding the graph Ramsey number , which is defined as the the minimum number  so that any -vertex graph contains either a copy of  or an independent set of size . For example, in 1947 Erdős pioneered the random coloring approach to obtain the lower bound , and in 1961 he developed the alteration method in order to obtain , see [7]. In 1975 and 1977 Spencer [26, 27] reproved these results via the Lovász Local Lemma, and also extended them to lower bounds on  for . In 1994 Krivelevich [18] further extended this to general graphs  via a new (large-deviation based) alteration approach, obtaining the lower bound

(1)

where the implicit constants may depend on  (writing  and , as usual). By analyzing (semi-random) -free processes, in 1995 Kim [17] and in 2010 Bohman–Keevash [3] have further improved the logarithmic factors in (1) for some graphs  such as triangles , cliques , and cycles . However, despite considerable effort, for  the best known lower and upper bounds are still polynomial factors apart, see [3, 4, 10]. Unsurprisingly, to further advance the proof methods, the field has thus stretched in several directions. One such widely-studied direction investigates online graph Ramsey games, with the goal of understanding what happens to various Ramsey numbers when decisions need to be made online.

In this note, we present a refinement of the above-mentioned widely-used alteration approaches of Erdős and Krivelevich (see e.g., [8, 17, 19, 20, 28, 12, 2, 13, 6, 22, 11]) that enables us to analyze online graph Ramsey games. As two concrete applications we consider Ramsey, Paper, Scissors games and online Ramsey numbers, each time extending recent bounds of Fox–He–Wigderson [11] and Conlon–Fox–Grinshpun–He [6].

1.1 Applications: Online Ramsey games

Our first application concerns the widely-studied online Ramsey game (see, e.g., [1, 21, 16, 5, 6]) that was introduced independently by Beck [1] and Kurek–Ruciński [21]. This is a game between two players, Builder and Painter, that starts with an infinite set  of isolated vertices. In each turn, Builder places an edge between two non-adjacent vertices from , and Painter immediately colors it either red or blue. The online Ramsey number  is defined as the smallest number of turns  that Builder needs to guarantee the existence of either a red copy of  or a blue copy of  (regardless of Painter’s strategy).

Our refined alteration approach enables us to prove a lower bound on  that, up to logarithmic factors, is about  times the best-known general lower bound for the usual Ramsey number , cf. (1).

Theorem 1 (Online Ramsey Game).

If  is a graph with , then as , where the implicit constant may depend on .

For general graphs , Theorem 1 gives the best known lower bounds for online Ramsey numbers. For -vertex cliques we obtain , which generalizes a recent bound of Conlon–Fox–Grinshpun–He [6, Theorem 4] for triangles, and also improves [6, Corollary 3] for small cliques. The best-known upper bounds  differ by a polynomial factor for , (see [6, Theorem 5]), analogous to the known gaps for . It would be interesting to investigate whether the lower bound of Theorem 1 can be improved if one replaces our alteration approach by an -free process [3] based approach or semi-random variants thereof [17, 13]; see also [6, Section 6].

Our second application concerns the fairly new Ramsey, Paper, Scissors game that was introduced by Fox–He–Wigderson [11]. For a graph , this is a game between two players, Proposer and Decider, that starts with a finite set  of  isolated vertices. In each turn, Proposer proposes a pair of non-adjacent vertices from , and Decider simultaneously decides whether or not to add it as an edge to the current graph (without knowing which pair is proposed). Proposer cannot propose vertex-pairs that would form a copy of  together the current graph, nor vertex-pairs that have been proposed before. The RPS number  is defined111For imperfect-information games such as Ramsey, Paper, Scissors (both players make simultaneous moves) one usually considers randomized strategies, see [23, pp. 16, 166], motivating why the definition of  includes probability of winning. as the largest number  for which Proposer can guarantee that, with probability at least  (regardless of Decider’s strategy), the final graph has an independent set of size .

Our refined alteration approach enables us to prove an upper bound on  for all strictly -balanced graphs , i.e., which satisfy  for all . This well-known class contains many graphs of interest, including cliques , cycles , complete multipartite graphs , and hypercubes .

Theorem 2 (Ramsey, Paper, Scissors Game).

If  is a strictly -balanced graph, then as , where the implicit constant may depend on .

For all strictly -balanced graphs , Theorem 2 gives the best known upper bounds for RPS numbers. For -vertex cliques we obtain , which generalizes the upper bound part of the very recent  result of Fox–He–Wigderson [11]. It would be interesting to obtain good (and perhaps again matching) lower bonds on  for other strictly -balanced graphs .

1.2 Main tool: Refined alteration approach

To motivate our refined alteration approach, we shall review related arguments for the Ramsey bound (1). Here Erdős [7] and Krivelevich [18] use a binomial random graph  with vertices and edge-probability to construct an -vertex graph  that (i) is -free and (ii) contains at least one edge in each -vertex subset , which implies . Standard Chernoff bounds suggest that the number  of edges of inside  is around , so for property (ii) it intuitively suffices to show that the alteration from  to  does not remove ‘too many’ edges from each -vertex subset .

To illustrate that this is a non-trivial task, let us consider the natural upper bound  on the number of removed edges from , where  denotes the collection of all -copies that have at least one edge inside . For any  it turns out that  due to ‘infamous’ upper tail [15, 24] behavior (see Appendix for the details). This lower bound not only rules out simple union bound arguments, but also suggests that one has to more carefully handle edges that are contained in multiple -copies.

For triangles , Erdős [7] overcame these difficulties in 1961 by a clever ad-hoc greedy alteration argument, showing that whp222In this note whp (with high probability) always means with probability tending to  as . the following works: If one sequentially traverses the edges of  in any order, only accepting edges that do not create a triangle together with previously accepted edges, then the resulting ‘accepted’ subgraph  satisfies (ii), and trivially (i). The fact that any edge-order works was exploited by Conlon et.al [6] and Fox et.al [11] in the analysis of triangle-based online Ramsey games.

To handle general graphs , Krivelevich [18] developed in 1995 an elegant alteration argument, showing that whp the following works: If one constructs  by deleting all edges that are in some maximal (under inclusion) collection  of edge-disjoint -copies in , then this (a) removes less than  edges from each -vertex subset , and (b) yields an -free graph by maximality of , establishing both (ii) and (i). Unfortunately, this slick maximality argument is hard to adapt to online Ramsey games, where players cannot foresee whether in future turns a given edge will be contained in an -copy or not.

Our refined alteration approach overcomes the above-discussed difficulties, by showing that whp the desired properties (i) and (ii) remain valid even if one deletes all edges from that are in some -copy (and not just some carefully chosen subset of these edges, as in the influential alteration approaches of Erdős and Krivelevich, cf. [7, 8, 18, 17, 19, 20, 28, 12, 2, 13, 6, 22, 11]). To state our main technical result, let  denote the number of edges in  that are in some -copy of . Recall that .

Theorem 3 (Main technical result).

Let  be a strictly -balanced graph. Then, for any , the following holds for all  and . Setting and , whp satisfies  for all -vertex sets .

Remark 4.

For any , the following holds for all  and . Setting  and  as in Theorem 3, whp satisfies  for all -vertex sets .

As discussed, our basic alteration idea is to construct by deleting all edges that are in some -copy of , so (i) holds trivially, and for suitable  then Theorem 3 and Remark 4 suggest that whp  for all -vertex subsets , establishing (ii). It is noteworthy that the largest independent sets of  (which have size less than ) are not much larger than those of , which are well-known to be of order  for  and thus , see [14, Section 7.1].

As we shall see in Section 2, variants of the above-discussed alteration argument carry over to certain online Ramsey games (where it will be useful that we can allow for arbitrary deletion of edges in -copies). We remark that the restriction to strictly -balanced graphs in Theorem 3 is often immaterial, since for (1) and related Ramsey bounds one can usually obtain the desired general bound by simply forbidding a strictly -balanced subgraph  with , cf. Section 2.2. Finally, in Section 4 we also discuss some further extensions of our alteration approach, including variants which forbid multiple hypergraphs.

1.3 Organization

In Section 2 we prove the discussed online Ramsey game results (Theorems 12) using the main technical result of our refined alteration approach (Theorem 3), which we subsequently prove in Section 3. Finally, in Section 4 we discuss some extensions of our alteration approach, including hypergraph variants.

2 Online Ramsey games

2.1 Ramsey, Paper, Scissors: Proof of Theorem 2

The following argument is based on a Decider strategy that randomly accepts edges (this strategy is completely oblivious, i.e., does not require knowledge of any proposed or accepted edges).

Proof of Theorem 2.

For  we choose  large enough and then  small enough so that Remark 4 and Theorem 3 both apply to  with and . We shall analyze the following strategy: in each turn Decider accepts the (unknown) proposed vertex-pair as an edge independently with probability . Let  denote the resulting final graph at the end of the game, i.e., which contains all accepted edges. Since all edges that do not create -copies are eventually proposed, there is a natural coupling between  and  which satisfies the following two properties: (a) that , and (b) that every edge in  is contained in an -copy of . Invoking Theorem 3 and Remark 4, it follows that this coupling satisfies the following whp: for any -vertex set  of  we have

which implies that the final graph  has whp no independent set of size . It follows that as  (where the implicit constant depends on ). ∎

2.2 Online Ramsey numbers: Proof of Theorem 1

The following argument is based on a Painter strategy that attempts to randomly color edges between high-degree vertices. The analysis is complicated by the fact that the game is played on an infinite set of vertices, which requires some care in the coupling and union bound arguments below.

Proof of Theorem 1.

For convenience we first suppose that  is strictly -balanced. For  we choose  large enough and then  small enough so that Theorem 3 applies to  with and . Set 

. At any moment of the game, we define 

as the set of all vertices that, in the current graph, are adjacent to at least  edges placed by builder (to clarify: the growing vertex set  is updated at the end of each turn).

We shall analyze the following strategy: Painter’s default color is blue, but if an edge  is placed inside , then Painter does the following independently with probability  (): it colors the edge  red, unless this would create a red -copy (), in which case the edge  is still colored blue. By construction there are no red -copies, and blue cliques  can only appear inside  (since all vertices in copy of  must be adjacent to at least  vertices). To prove  as  (with implicit constants depending on ), by the usual reasoning it remains to show that after  steps there are whp no blue cliques  inside . Let  denote the collection of all -vertex sets  after  steps. Intuitively, the plan is to show that, inside each vertex set  that can become a blue clique , there are more red-coloring attempts () than ‘discarded’ red-coloring attempts (), which enforces a red edge inside .

Turning to details, note that  during the first  steps. Using the order in which vertices enter  (breaking ties using lexicographic order), at any moment during the first  steps we thus obtain an injection . After  steps, we abbreviate this injection by , and write . Define  as the event that, during the first  steps, the number of ‘discarded’ red-coloring attempts () inside  is at most . There is a natural turn-by-turn inductive coupling between  and Painter’s strategy, where the red-coloring attempt () occurs if  is an edge of . A moments thought reveals that, during the first  steps, under this coupling the total number of ‘ignored’ red-colorings () inside  is at most  defined with respect to  (since () can only happen when a red-coloring of  creates a red -copy, which under the coupling implies that  is contained in an -copy of ). Applying Theorem 3 with  to , using the described coupling and  it then follows that, whp, the event  occurs for all .

Intuitively, we shall next show that, for all -vertex sets  that contain  edges (a prerequisite for having a blue clique  inside ), the number of red-coloring attempts () inside  is at most . To make this precise, define  as the event that builder places less than  edges inside  during the first  steps. Let  denote the number of red-coloring attempts () inside  during the first  steps, and define  as the event that . Let  denote the collection of all -vertex sets . Since  defines an injection from  to , writing  it follows that

(2)

Fix , and set . Note that, by checking in each turn for red-coloring attempts () inside , we can determine  without knowing  in advance. Furthermore, since every vertex is adjacent to at most  vertices before entering , the event  implies that during the first  steps at least  red-coloring attempts () happen inside , each of which is (conditional on the history) successful with probability . It follows that 

stochastically dominates a binomial random variable 

, unless the event  occurs. Noting  and , by invoking standard Chernoff bounds (see, e.g., [14, Theorem 2.1]) it then follows that

(3)

Combining (2)–(3) with , we readily infer that, whp, the event  occurs for all .

To sum up, the following holds whp after  steps: every -vertex set  contains either (a) at least  red edges, or (b) less than  edges in total. Both possibilities prevent a blue clique  inside , and so the desired lower bound  follows (as discussed above).

Finally, in the remaining case where  is not strictly -balanced, we pick a minimal subgraph  with . It is straightforward to check that, by construction,  is strictly -balanced. Furthermore, since any -free graph is also -free, we also have . Repeating the above proof with  replaced by  then gives the claimed lower bound on . ∎

3 Refined alteration approach

3.1 Bounding : Proof of Theorem 3

For Theorem 3 the core strategy is to approximate  by more tractable auxiliary random variables, inspired by ideas from [15, 31, 30, 25]. In particular, we expect that the main contribution to  should come from -copies that share exactly two vertices and one edge with ; in the below proof we denote the collection of such ‘good’ -copies by . Note that when multiple good -copies from  contain some common edge  inside , they together only contribute one edge to . It follows that, by arbitrarily selecting one ‘representative’ copy  for each relevant edge , we should obtain a sub-collection  of good -copies with . The -copies in  share no edges inside  by construction, and it turns out that all other types of edge-overlaps are ‘rare’, i.e., make a negligible contribution to . We thus expect that there is an edge-disjoint sub-collection  of good -copies with , and here the crux is that the upper tail of 

is much easier to estimate than the upper tail of 

(see Claim 6

 below). The following proof implements a rigorous variant of the above-discussed heuristic ideas.

Proof of Theorem 3.

Noting that the claimed bounds are trivial when  (since then there are no -vertex sets  in  due to ), we may henceforth assume .

Fix a -vertex set . Let  denote the collection of all -copies in  that have at least one edge inside , and let  denote the sub-collection of -copies that moreover share exactly two vertices with . Let  denote a size-maximal collection of edge-disjoint . Clearly , and Claim 5 below establishes a related upper bound. Let  denote a size-maximal collection of edge-disjoint . Let  denote a size-maximal collection of edge-disjoint  with distinct that satisfy  and . Let  denote the number of -copies in  that contain the edge , and define  as the maximum of  over all .

Claim 5.

We have .

Proof of Claim 5.

We divide the -copies in  into two disjoint groups: those which share at least one edge with some  or , and those which do not; we denote these two groups by  and , respectively. For , let  denote the collection of edges from  that are contained in at least one -copy from . Note that  and . Turning to , by maximality of  and  we infer the following two properties of : (a) all -copies intersect with  in exactly two vertices, so , and (b) any two distinct -copies are edge-disjoint, unless they both intersect  in the same two vertices. For each we now arbitrarily select one -copy from  that contains . By properties (a)–(b) of  and size-maximality of , this yields a sub-collection  of edge-disjoint -copies satisfying , and the claim follows. ∎

The remaining upper tail bounds for , , and  hinge on the following four key estimates. First, and strictly -balancedness of  imply , so that

(4)

Second, and  imply that there is  such that

(5)

Third, using  and strictly -balancedness of  (implying that  for all  with ), it follows that there is  such that

(6)

The below-claimed fourth estimate can be traced back to Erdős and Tetali [9]; we include an elementary proof for self-containedness (see [31, Section 2] for related estimates that also allow for overlapping edge-sets).

Claim 6.

Let  be a collection of edge-subsets from . Define  as the largest number of disjoint edge-sets from  that are present in . Then  for all .

Proof of Claim 6.

Set . Exploiting edge-disjointness and , it follows that

which completes the proof by noting that the function  is decreasing for positive . ∎

We are now ready to bound the probability that  is large. Since  is strictly -balanced, it contains no isolated vertices and thus is uniquely determined by its edge-set. This enables us to apply Claim 6 to  (as  is a size-maximal collection of edge-disjoint -copies from ). Using estimate (4), it is routine to see that, for , the associated parameter  from Claim 6 satisfies

(7)

Noting  and , now Claim 6 (with ) implies that, for ,

(8)

Next, we similarly use Claim 6 to bound the probability that  is large. For the associated parameter  we shall proceed similar to (7) above: using estimates (4)–(5), for  we obtain

(9)

With similar considerations as for (8) above, for  Claim 6 (with ) then yields

(10)

We shall analogously use Claim 6 to bound the probability that  is large. For the associated parameter , the basic idea is to distinguish all possible subgraphs  in which the relevant  can intersect. Also taking into account the number of vertices which  and  have inside , i.e., , by definition of  it now follows via estimates (4)–(6) that

(11)

(To clarify: in (11) above we used that (6) implies  for all  with .) Similarly to inequalities (8) and (10), for  now Claim 6 (with ) yields

(12)

Finally, combining (8), (10) and (12) with Claim 5, a standard union bound argument gives

(13)

To complete the proof of (13), it thus remains to show that, for , we have

(14)

Using (4), (6) and , this upper tail estimate for  follows routinely from standard concentration inequalities such as [30, Theorem 32], but we include an elementary proof for self-containedness (based on ideas from [29, 31]). Turning to the proof of (14), let  denote the number of -copies in  that contain the edges , and define  as the maximum of  over all distinct . We call an -tuple  of -copies an -star if each  contains the edges  and satisfies . Define  as the number of -stars  that are present in . Summing over all -stars , by noting that the intersection of with is isomorphic to some proper subgraph  containing at least  edges, using estimates (4) and (6) it then is routine to see that, for , we have

Since trivially , using  we infer . Consider a maximal length -star  in , and note that in  any -copy containing the edges  is completely contained in  (by length maximality), so that  holds (using that  is uniquely determined by its edge-set). For  it follows that

(15)

With an eye on , let  denote the collection of all -copies in  that contain the edge . We pick a subset  of -copies in  that is size-maximal subject to the restriction that all -copies are edge-disjoint after removing the common edge . For any , note that in  there are a total of at most  copies of  that share  and at least one additional edge with . Hence  and  imply  for  (by maximality of ). As the union of all -copies in  contains exactly edges, using  and  it follows that

(16)

Using estimate (4), for  the right-hand side of (16) is at most . Recalling , by taking a union bound over all edges  it then follows that

(17)

which together with (15) completes the proof of estimate (14) and thus Theorem 3. ∎

The above proof of (14) can easily be sharpened to for suitable , see (16)–(17). Together with the proof of (13) and , this implies that, whp,  for all -vertex sets , which intuitively suggests that  is well-approximated by .

3.2 Bounding : Proof of Remark 4

Remark 4 follows easily from Chernoff bounds; we include the routine details for completeness.

Proof of Remark 4.

Noting  and , by invoking standard Chernoff bounds (see, e.g., [14, Theorem 2.1]) it follows, for  large enough, that