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 . 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  further extended this to general graphs via a new (large-deviation based) alteration approach, obtaining the lower bound
where the implicit constants may depend on (writing and , as usual). By analyzing (semi-random) -free processes, in 1995 Kim  and in 2010 Bohman–Keevash  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  and Conlon–Fox–Grinshpun–He .
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  and Kurek–Ruciński . 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  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 . 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 . 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  and Krivelevich  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  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  and Fox et.al  in the analysis of triangle-based online Ramsey games.
To handle general graphs , Krivelevich  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 .
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.
In Section 2 we prove the discussed online Ramsey game results (Theorems 1–2) 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 defineas 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
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
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 .
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
Second, and imply that there is such that
Third, using and strictly -balancedness of (implying that for all with ), it follows that there is such that
The below-claimed fourth estimate can be traced back to Erdős and Tetali ; we include an elementary proof for self-containedness (see [31, Section 2] for related estimates that also allow for overlapping edge-sets).
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
Noting and , now Claim 6 (with ) implies that, for ,
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
To complete the proof of (13), it thus remains to show that, for , we have
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
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
3.2 Bounding : Proof of Remark 4
Remark 4 follows easily from Chernoff bounds; we include the routine details for completeness.