The bipartite $K_{2,2}$-free process and bipartite Ramsey number $b(2, t)$

The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from $X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and show that the resulting graph witnesses that $b(2,t) =\Omega \left(t^{3/2}/\log t \right)$, thereby improving the best known lower bound.


Introduction
The bipartite Ramsey number b(s, t) is the smallest integer n such that every blue-red edge coloring of K n,n contains either a blue K s,s or a red K t,t . This definition was first introduced by Beineke and Schwenk [2] in 1976. We will find it convenient to define the bipartite independence number of a graph G ⊆ K n,n as the largest value of t such that there are sets of vertices A and B on opposite sides of the bipartition of K n,n such that |A| = |B| = t and G has no edges in A × B. Thus, b(s, t) > n if and only if there exists a K s,s -free graph G ⊆ K n,n that has bipartite independence number less than t. The best known lower and upper bounds on the diagonal problem, due to Hattingh and Henning [13] and Conlon [9] respectively, are √ 2 e t2 t/2 b(t, t) (1 + o(1))2 t+1 log 2 t.
As is the case for the ordinary Ramsey number, there is an exponential gap which remains to be closed. In this note, we are concerned with the simplest nontrivial "off-diagonal" case, b(2, t). The best known lower and upper bounds, both due to Caro and Rousseau [8], are The upper bound follows directly from the well known upper bounds on the Zarankiewicz problem. z(n, s) is the largest number of edges in a K s,s -free subgraph of K n,n . A theorem of Kövari, Sós and Turán [15] says that z(n, s) = O(n 2−1/s ). By the pigeonhole principle, if the number of edges in K n,n exceeds z(n, 2) + z(n, t), then in any blue-red coloring of the edges, one color class must exceed its respective Zarankiewicz bound. Thus solving the inequality n 2 > c 1 n 3/2 + c 2 n 2−1/t for n provides the upper bound. The proof of the lower bound in [8] makes use of the Lovász Local Lemma. In this paper we improve their lower bound by a logarithmic factor: Theorem 1. We have the following lower bound on b(2, t): b(2, t) Ω t 3/2 log t .
Let R(G 1 , G 2 ) represent the ordinary two color Ramsey number, i.e., the smallest integer n such that every blue-red edge coloring of the edges of K n contains a blue copy of G 1 or a red copy of G 2 . When G 1 = K s , G 2 = K t , we write R(s, t). The search for the asymptotics of the off diagonal ordinary Ramsey number R(3, t) has a long and interesting history as laid out by Joel Spencer in [20]. Currently, the best known results differ only by a constant factor. Shearer [19], improving a result of Ajtai, Komlós and Szemerédi [1], proved that R(3, t) (1 + o(1))t 2 / log t. Kim [14] first proved that R(3, t) Ω(t 2 / log t), and Bohman [4] reproved the same bound using a random process first considered by Erdős, Suen and Winkler [10] and now commonly referred to as the triangle-free process. This is a stochastic process in which random edges are added to an empty graph one by one under the constraint that no triangles are formed. (Kim's original proof in [14] considered a "semi-random" version where we receive many random edges at a time instead of one at a time.) Recently, Bohman and Keevash [6], and independently Fiz Pontiveros, Griffiths and Morris [11], improved the analysis of the triangle-free process, and their analysis implies that R(3, t) ( 1 4 − o(1))t 2 / log t. Of course one can also consider the H-free process for any H, where edges are randomly added one by one under the constraint that no copy of H is formed. Ruciński and Wormald [18] were among the first to consider such a process, analyzing the d-process, which is the H-free process where H is a star on d + 1 vertices. Further work has been done to analyze the H-free process for other families of graphs H, mostly when H is a clique or a cycle (see for example Picollelli [16,17] and Warnke [21,22]). Bohman and Keevash [5] have the most general results for the H-free process; they analyze the process and bound the independence number of the resulting graph for a large class of graphs H including cycles of any length as well as cliques of any size (but also all strictly 2-balanced graphs), establishing new lower bounds on Ramsey numbers R(H, K t ) where H is any fixed cycle or clique and t → ∞. Bohman, Picollelli and Mubayi [7] studied the H-free process for certain hypergraphs H, resulting in new lower bounds for the corresponding hypergraph Ramsey numbers.
Inspired by the previous work on H-free processes, we study the bipartite K 2,2 -free process, a version of the H-free process in a large balanced bipartite host graph (as opposed to the standard H-free process which uses K n as a host graph). The process begins with an empty graph G 0 on vertex set X ∪ Y where |X| = |Y | = n. We form the graph G i by adding to G i−1 an edge e i chosen uniformly at random from all pairs of vertices in X × Y which do not already appear in G i−1 and which do not create a copy of K 2,2 . Let M be the random variable representing the number of edges in the final graph produced at the end of the process. Then G M is K 2,2 -free by construction. The main contribution of this paper is to prove that with high probability, G M has bipartite independence number at most Cn 2/3 log 2/3 n for some constant C.
Theorem 2. With high probability, the graph produced at the end of the bipartite K 2,2 -free process has bipartite independence number O(n 2/3 log 2/3 n).
In the next section we prove Theorem 2, from which Theorem 1 follows trivially.

Proof of Theorem 2
Bennett and Bohman [3] proved a general result which we will find useful. We build our proof "on top of" the proof of Theorem 1.1 in [3], in the sense that we will use not only the statement of that theorem but also some other facts that are established in its proof. In particular, Theorem 1.1 in [3] is proved by establishing dynamic concentration of a family of random variables. In our proof, we will use (without further justification) the fact that these variables are dynamically concentrated. One of the goals of this paper is to demonstrate the utility of [3] as a "black box" that takes care of a lot of the work of analyzing these processes, making for shorter proofs.
In Section 2.1 we will summarize the relevant results from [3]. In Section 2.2 we prove a few lemmas which do not follow directly from the results in [3], namely, bounds on the maximum degree and on the maximum density of subsets. Finally, in Section 2.3 we bound the bipartite independence number of the graph produced by the process by using similar proof techniques to those used in [5] for analyzing the H-free process when H is a cycle.

The Black Box
In this section, we summarize the results from Bennett and Bohman [3] which we will utilize. Let H be a hypergraph on vertex set V (i.e. H is a collection of subsets of V and the sets in this collection are the edges of H). An independent set in H is a set I ⊆ V such that I contains no edge of H. The random greedy independent set process (or just the independent process) forms a maximal independent set in H by iteratively choosing vertices at random to be in the independent set. To be precise, we begin with H(0) = H, V (0) = V and I(0) = ∅. Given independent set I(i) and hypergraph H(i) on vertex set V (i), a vertex v ∈ V (i) is chosen uniformly at random and added to I(i) to form I(i + 1). The new vertex set V (i + 1) and new hypergraph H(i + 1) are formed by 1. removing v from every edge in H(i) that contains v (so these edges become smaller edges), 2. deleting v from V (i), and 3. deleting from V (i) every vertex that is in a singleton edge (and any such edge containing a deleted vertex is removed).
Define the degree of a set A ⊆ V to be the number of edges of H that contain A. For a = 2, . . . , r − 1 we define ∆ a (H) to be the maximum degree of A over A ∈ V a . We also define the b-codegree of a pair of distinct vertices v, v to be the number of pairs of edges e, e ∈ H such that v ∈ e \ e , v ∈ e \ e and |e ∩ e | = b. We let Γ b (H) be the maximum b-codegree of H.
and Γ r−1 (H) < D 1−ε then the random greedy independent set algorithm produces an independent set I in H with The K 2,2 -free process in the host graph K n,n is an instance of the independent process in a hypergraph H K 2,2 = H K 2,2 (n), where each vertex in H K 2,2 corresponds to an edge in K n,n and edges in H K 2,2 correspond to sets of edges in K n,n that form copies of K 2,2 . Thus H K 2,2 is a 4-uniform, (n − 1) 2 -regular hypergraph on n 2 vertices. In the notation of Theorem 3, we have r = 4, so the conditions of Theorem 3 are met with r = 4 and any 0 < ε < 1/3. Thus, Theorem 3 tells us that the bipartite K 2,2 -free process admits Ω n 4/3 log 1/3 n many edges. For the rest of the paper, we will almost exclusively use language referring to the K 2,2 -free process as opposed to language referring to the independent process on H K 2,2 . Now we state the dynamic concentration results that are established in [3] as part of the proof of Theorem 3. As in [3] we define the scaled time parameter In the proof of their theorem, Bennett and Bohman establish dynamic concentration of the random variables Q and d 2 (e) (see equations (8) and (9) in [3]). In particular, their proof implies that there exists a positive constant ε such that w.h.p.
for all i i max := εn 4/3 log 1/3 n. Note that lines (3) and (4) significantly simplify the bounds in [3] which actually depend on several constants. However it is not difficult to see that the bounds given in [3] imply that (3) and (4) hold for some sufficiently small ε > 0. In this paper we will always assume that n is sufficiently large and ε > 0 is sufficiently small, and will write some inequalities that only hold under those assumptions.

High degrees and dense subgraphs
Let G i be the K 2,2 -free graph at step i. In this section we show that w.h.p. G i does not have any vertices of degree too high, nor subgraphs that are too dense. More specifically, G i resembles (at least in these aspects) a binomial random bipartite graph with edge probability i n 2 . Let E i be the event that (3), (4) hold for all steps up to and including step i, and recall that P(E imax ) = 1 − o(1).

Lemma 4.
For any set of edges F ⊆ E(K n,n ), we have Proof. The probability that all edges of F are chosen is at most the number of ways to specify which steps these edges will be chosen multiplied by the probability of choosing the prescribed edges in the specified steps. If we know that E imax holds,then for any i i max and so the probability of choosing a particular edge on a particular step is at most 1 Q < 1+o(1) n 2−ε 3 (conditional on the history of the process). So we have that the probability that all edges of F are chosen is at most Proof. By Lemma 4, we have that the probability a particular vertex has degree at least x at step i max (and hence at any step in the process) is at most Thus, if we let Z represent the number of vertices with degree at least n 1/3+3ε 3 , then The result follows from Markov's inequality. On the second and fourth line we have used the fact that h 2ε −3 (a + b) and on the second line we used h 2ε −3 abn −2/3+3ε 3 .

Bipartite Independence Number
Let I X ⊆ X and I Y ⊆ Y with |I X | = |I Y | = α := 2ε −1 n 2/3 log 2/3 n. We will track Q I by writing Q I = Q I − A I , where A I represents the effect of "large" one-step changes. Formally, define and Q I = Q I + A I .
Our motivation for defining A I is to ensure that the one-step change of Q I is not too large. We will then be able to apply a martingale inequality. We will establish dynamic concentration for Q I , and just a crude bound for A I which can be regarded as an error term.

Bounding A I
In this section, we prove the following lemma which provides an upper bound on A I .

Lemma 7.
With high probability, for every Proof. Assume that E imax and (5), (6) hold. Under these assumptions, we will show that for every I and every i i max , A I (i) n 1+45ε 3 . Fix an I = I X × I Y . The sets I (1) Y which we define below are all dependent on i and the graph G i , but as usual, we omit this from our notation. Let I (1) X ⊆ Y be the set of vertices with at least n 1/3−16ε 3 neighbors in I X (in graph G i ), and similarly define I Recall that |I X | = α = 2ε −1 n 2/3 log 2/3 n. Then by (6) we have

|I
(1) If the maximum above were achieved by the second argument, we would have a contradiction since n 3ε 3 log 2/3 n n 1/3−16ε 3 . Thus we have |I (1) By (6), we have

|I
(2) The first argument must be the maximum otherwise we get a contradiction. Thus using (8), we have |I (2) X |) and so rearranging (and since I Note that by (6) and (8) we have w.h.p., since the maximum is the first argument. By (5) and (9) we have e(I X , I Y )+e(I X , I Y ) O n 34ε 3 · n 1/3+3ε 3 = O n 1/3+37ε 3 w.h.p.. Thus we have proved (7).
In order to have |∆Q I (i)| > n 2/3−12ε 3 the edge (x i , y i ) chosen at step i must be in one of the following three sets: I S (v) be the number of vertices in S that are distance 2 from v. Then and since (x i , y i ) / ∈ I X × I Y , we can bound d Y . Then y i has at most n 1/3−16ε 3 neighbors in I (1) Y which each have at most n 1/3+3ε 3 neighbors in I Y . y i has at most n 1/3+3ε 3 additional neighbors that all have at most n 1/3−16ε 3 neighbors in I Y . Thus the electronic journal of combinatorics 27(4) (2020), #P4.23 We have the same bound on d (2) I (x i )1 y i ∈I Y by symmetry. To bound d I (x i )d I (y i ), note that at least one of Thus using (7), we see that the number of steps with |∆Q I | n 2/3−12ε 3 can be bounded by e(I where we use the fact that the largest possible value of |∆Q I (i)| is the square of the maximum degree, n 1/3+3ε 3 .

Dynamic concentration of Q I (i)
Define E i ⊆ E i as the event that E i holds, that (5) and (6) hold, and that for all j i we have where f (t) = n 4/3−5ε 3 e t 3 +t .
By Lemmas 5 and 6 we have that (5) and (6) hold w.h.p. and in the last section we proved that in the event that (5) and (6) hold we (deterministically) have for all I and for all j i that A I (j) < n 1+45ε 3 .
Note that since t ε log 1/3 n we have f (t) n 4/3−3ε 3 . We now define Q I We will show that Q I + is a supermartingale. Since ∆ Q I + (i) = 0 if E i fails to hold, we will assume E i holds.
Recall that d 2 (uv) is the number of potential copies of K 2,2 containing the edge uv in which two edges of the K 2,2 other than uv are in G(i) and the last pair is open. By (3) and (4), we know that Q ∈ (1 ± n −10ε 3 )n 2 q and d 2 (u, v) ∈ (1 ± n −10ε 3 )3n 2/3 t 2 q where we recall that q = q(t) = e −t 3 . We first calculate and since by (8) and (9) we have the electronic journal of combinatorics 27(4) (2020), #P4.23 Now we use Taylor's theorem to bound the one-step change of α 2 q(t) + f (t), the deterministic terms in Q I + . We have where on the second line when we write O (α 2 q (t) + f (t)) 1 n 8/3 we mean an absolute bound on the function inside the big-O that holds holds for all t ε log 1/3 n. Considering the particular functions q(t), f (t) we arrive at the big-O term on the last line. Now we will do the supermartingale calculation for Q I + . Throughout the following, keep in mind that α 2 3t 2 qn −4/3 n 2ε 3 , and that Thus, using (12), (13), and the fact that we have dynamic concentration in the event E i we have that In the third line, we have used the geometric series expansion 1/(1+n −10ε 3 ) = 1−n −10ε 3 + O(n −20ε 3 ) to calculate 1 − n 1+45ε 3 +f where we used that n 1+45ε 3 α 2 q n −20ε 3 . Thus Q I + is a supermartingale. We will use the following martingale inequality found in [12]. .
In order to apply Lemma 8, we must bound the variance.
In a completely analogous fashion, we may prove that Q I − remains non-negative until time i max for every I, w.h.p. Thus E imax holds w.h.p..

Final bound on bipartite independence number
Now for any fixed I we will show that it is very likely that some edge is chosen in I. We have already shown that the event E imax holds w.h.p., so in particular we have that Q = n 2 q(1 + o(1)) and Q I = α 2 q(1 + o(1)) for all i i max . Thus the probability of choosing an edge in I at any given step is Q I /Q = (1 + o(1)) α 2 q n 2 q 3ε −2 n −2/3 log 4/3 n. So the probability that no edge from I is chosen in the first i max steps is at most 1 − 3ε −2 n −2/3 log 4/3 n εn 4/3 log 1/3 n exp −3ε −1 n 2/3 log 5/3 n .
which is small enough to overcome a union bound over (15) many choices for the set I.

Conclusion
In [8], Caro and Rousseau write, "our knowledge of b(2, n) closely parallels that of R(C 4 , K n )." In this paper, we have furthered this parallel, but the most intriguing open question (as mentioned in [8]) is to prove or disprove that b(2, t) = o(t 2−ε ) for some ε > 0. Erdős famously conjectured that R(C 4 , K n ) = o(n 2−ε ). It may be the case that improving the upper bound on b(2, t) is a simpler task than improving that of R(C 4 , K n ). Another direction would be to extend our technique to find new lower bounds on b(s, t) with s fixed and t → ∞ or on the bipartite Ramsey number of a fixed cycle versus a large bipartite clique. The techniques used in this paper could be applied here, but we have opted not to pursue this for the sake of brevity. Most likely the techniques in [17,21] can also be used to show that the bipartite K 2,2 -free process terminates in O(n 4/3 log 1/3 n) steps, matching the lower bound we proved in this paper.