Finding unavoidable colorful patterns in multicolored graphs

Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollob\'as conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Mont\'agh. Here, we obtain a characterization for arbitrary values of $r$, discuss quantitative bounds, as well as generalizations to infinite graphs.


Introduction
Graph Ramsey Theory refers to mathematical results that attempt to find large patterns in colored graphs. The pattern in question is most often a monochromatic clique, a complete subgraph whose edges are all the same color. The Ramsey number R(k) is the smallest integer n for which every 2-edge-coloring of the complete graph K n contains a monochromatic clique on k vertices. It is known that 2 k/2 ≤ R(k) ≤ 2 2k , where the constants that appear in the exponents resisted improvements for decades. For an overview, we refer the reader to the 2015 survey by Conlon et al. [1].
To find patterns that are not monochromatic, we need to assume that all color classes are sufficiently represented in the colorings we consider. Definition 1.1. We call χ : E(K n ) → [r] ε-balanced if each color class has at least ε n 2 edges. When it is clear from context, we call K n ε-balanced if it comes equipped with an ε-balanced coloring.
A result by Erdős and Szemerédi shows that graphs which are not ε-balanced contain monochromatic cliques larger than the general bounds would be able to provide [4]. So in some sense, even if we were not interested in finding bi-colored patterns, ε-balanced graphs are the natural graphs to study from a Ramsey theoretic standpoint.
The natural function to look at is the ε-balanced Ramsey number, which we introduce below. Note that by a color-consistent copy, we mean a subgraph which also preserves the color structure up to relabelling of the colors. Definition 1.2. Let r ∈ N, fix some ε with 0 < ε < 1/r, let H be some graph with an associated r-edge-coloring. We denote by R r ε (H) the smallest N ∈ N such that for any n ≥ N , if χ ε is some ε-balanced r-edge-coloring of K n , then K n contains a color-consistent copy of H. If no such N ∈ N exists, we say R r ε (H) = ∞. When r is not specified, R ε (H) denotes R 2 ε (H). Also, if H is a family of colored graphs, R r ε (H) denotes the smallest positive integer N for which for all n ≥ N , all ε-balanced K n contain a colorconsistent copy of some element in H. Figure 1: Elements of F 3 . Theorem 1.4 states that a large enough 3-edge-colored K n will necessarily contain a "blown-up" version of at least one of these graphs.
Let A be the graph with two red cliques of size k adjacent entirely in blue, and B be the graph with a red clique and a blue clique of size k adjacent entirely in blue. Let I k := {A, B}. Since A and B are balanced, the only H for which R 2 ε (H) is finite are H which appear in both elements of I k .
In the other direction, Bollobás conjectured that R 2 ε (I k ) is finite, and this was proved by Cutler and Montagh, who showed R 2 ε (I k ) ≤ 4 k/ε [2]. Fox and Sudakov later improved the bound, showing R 2 ε (I k ) ≤ (16/ε) 2k , which is asymptotically tight [6]. In particular, this result completely characterizes the 2-edge-colored H for which R 2 ε (H) is finite. A natural question is to similarly classify the r-edge-colored H for which R r ε (H) is finite. Our first result is in this direction. We first introduce some terminology. By a fully-colored graph, we mean a graph which has a coloring of its vertex set as well as its edge set. If G is a fully-colored graph, by the t-blow-up of G we denote the edge-colored graph G t where the colored vertices are replaced with monochromatic cliques of size t, and the colored edges are replaced with monochromatic complete bipartite graphs in their respective colors. If F is a family of fully-colored graphs, F t := F ∈F F t . We now define a class of families which will take the role of I k in a multicolored setting, in that this new family will characterize H for which R r ε (H) < ∞. Definition 1.3. Let r ≥ 2 be an integer. By F r we denote the unique (up to permutation of the colors within each graph) maximal family of r-fully-colored complete graphs with the following properties: (1) For all F ∈ F r , F uses all r colors (either via vertices or edges) (2) If any vertex v from any F ∈ F r is removed, F \ {v} does not use all r colors.
(3) No two F 1 , F 2 ∈ F r are color-consistent copies of each other.
It is easy to see that there are no graphs on more than 2r − 2 vertices which satisfy both (1) and (2). So in particular, F r is always finite. Also, for every r there is an ε r such that all elements of F r are ε r -balanced. (3) only serves an aesthetic function, we do not want any redundancies in F r .
For example, F 2 consists of a graph with 2 red vertices and a blue edge between, and a graph with a red and a blue vertex and a blue edge between. This generalizes the earlier family we introduced, as I k is the same as F 2 k . See Figure 1 for a representation of F 3 . We have obtained the following generalization of Bollobás's conjecture: Theorem 1.4. For any r ∈ N and ε with 0 < ε < 1/r, there exists a constant c := c(r) such that for any k ∈ N, we have that R r The result is asymptotically tight by a simple probabilistic construction. Further, the result characterizes r-edge-colored H with a finite value of R k ε (H) as those which are color-consistent subgraphs of all elements of F r whenever ε is small enough with respect to r. The technicality is that even if we allowed ourselves to have parts with different sizes, not all elements of F r can be made ε-balanced, if ε is large (for example, if ε = 1/r). For example, observe that the unique element of F 3 with 2 vertices can never be blown up to be ε-balanced if ε > 1/4, hence H with R 3 ε (H) < ∞ need not be a color-consistent subgraph of all elements of F 3 when ε is large. Even though Theorem 1.4 is best possible asymptotically in general, we manage to get better upper bounds for R 2 ε (H) for certain "asymmetric" H. The pattern defined below is a weakening of F 2 k . Definition 1.5. By M l,k , we denote a family of 2-edge-colored graphs with vertex set V := L R where |L| = l and |R| = k, and R is a (WLOG) red clique, L × R is a blue complete bipartite graph, and L is an independent set.
See Figure 2 for clarity. Observe that M 0,k is simply a homogeneous set, M 1,k is a particular coloring of a complete graph. The below result shows that when l is treated as a constant, R 2 ε (M l,k ) is at most a constant factor away from R(k). Theorem 1.6. For any 0 < ε < 0.5, and l ∈ N, there exists a C := C(ε, l) such that R ε (M l,k ) ≤ C · R(k).
This gives a new proof of Bollobás's conjecture as M R(k),k necessarily contains an element of F 2 k . The bound we obtain here is worse in general, but has better dependence on k. If we use the technique of our Theorem 1.4 or the Fox-Sudakov result, the bound would have to have a multiplicative dependence on k of the form ε −ck for some positive constant c. In contrast, R(k) ≤ 2 2k , with no dependence on ε.
We are also interested in explicit values of ε-balanced Ramsey numbers for small patterns. Here, it is natural to fix ε at 0.5. The simplest case where the explicit value of R 2 0.5 (M l,k ) is non-trivial is when l = 1 and k = 3. The Paley graph on 9 vertices does not contain an M 1,3 , and from the other side one can use our methods to obtain: We believe it should be possible to close this large gap. Recall that R(4) = 18, and the unique construction on 17 vertices without a 4-clique is a Paley graph.
We also consider some infinite analogues of these questions. For graphs defined on the naturals, i.e. [N] 2 = {A ⊆ N : |A| = 2}, an easy coloring shows that the analogue of Bollobás's conjecture does not hold even if we assume every vertex is back degree balanced (i.e. half of the edges from vertex i to vertices in [i − 1] are blue and half red) and has infinite degree in both colors. However for graphs defined on R, we see that non-meagreness corresponds well with our notion of ε-balancedness in the finite case.
be a coloring such that each color class is non-meagre and has the Baire property (as a subset of R 2 ). Then f yields some color-consistent copy of F r c . The proof techniques for Theorems 1.4 and 1.8 are very similar, except topological lemmas such as the Localization lemma take the place of combinatorial lemmas such at the Dependent Random Choice lemma. We give the necessary definitions in Section 4.

Proof of Theorem 1.4
We start with the simple observation that if we can find a complete subgraph H which is the disjoint union of monochromatic cliques of size r, the edge set between any two such monochromatic clique is also monochromatic (call H "fully-complete multipartite" if it is of this form), and further, all k colors appear somewhere in H, then H must contain an element of F k r . We should note that if one is not interested in asymptotically tight bounds, getting a fullycomplete multipartite graph which uses all r colors is not difficult. Indeed, a celebrated theorem by Kövari, Sós and Turán states that if a color class has an ε-fraction of the edge set, then we can find a monochromatic subgraph K s,s where s = c ε log n and c ε > 0 depends only on ε. Invoking the Kövari-Sós-Turán theorem on each of the r color classes in an ε-balanced K n , we obtain already obtain a subgraph that uses all r colors. Now, we can apply Ramsey's theorem to all 2r parts to obtain monochromatic cliques, and afterwards simply use that between any two subset of the vertices, at least one color class has 1/r-fraction of the edge set, and iteratively invoke the Kövari-Sós-Turán theorem to get a smaller subgraph where the edge-set between all 2r monochromatic cliques is also monochromatic. This already proves: Proposition 2.1. For any r ∈ N and ε with 0 < ε < 1/r, R r ε (F r k ) ≤ ∞ In fact, since all elements of F r k are ε r -balanced for some ε r < 1/(4r 2 ) (as any element of F r k has at most 2r − 2 vertices), we can state: When r = 2, the argument we have sketched above seems to give the simplest proof of Bollobás's conjecture. However, for arbitrary r, it gives a tower-type bound where the height of the tower is Θ(r). On the other hand, our Theorem 1.4, combined with a simple probabilistic construction, will establish that the correct order of magnitude of R r ε (F r k ) at ε −ck , where c is some positive constant that depends only on r.
However, the layout of the better proof will be very similar in spirit to the simpler argument we provided above. In particular, we will again aim to find a fully-complete multipartite subgraph that uses all r colors. The trick will be to use the dependent random choice technique, which recently proved to be a powerful tool in combinatorics. We refer the reader to the survey by Fox and Sudakov for an overview [5]. We should also note that Fox and Sudakov also used the dependent random choice technique to obtain the asymptotically sharp bound on R r ε (F r k ) when r = 2, in [6]. The following lemma is the most basic form of the technique, a proof can be found in [5]: Let G be a graph with average degree εn. Then, there exists a subset W with |W | = w such that all k 0 sized subsets of W have a common neighborhood of size βn, provided that there exists a positive integer t satisfying: We will need a bipartite version of this Lemma. The proof, which is very similar to the proof in [5], is also provided for completeness.
Lemma 2.4. Let G be a graph partitioned into vertex sets A and B, with |A| = m and |B| = m , and at least εmm edges between A and B. Then, there exists W ⊆ A, with |W | = w, such that every subset K ⊂ W with |K| = k 0 has βm common neighbors in B, provided that there exists a positive integer t satisfying: Using this fact and linearity of expectation, we calculate: where the inequality holds by convexity of x t . Now, if we remove from U an element from every k 0 -element subset R which fails to have |N B (R)| ≥ βm , the resulting set will have the desired properties. Let Z be the random variable denoting how many such k 0 -element subsets are there in U .
Thus, on average, we would have to remove at most m k0 β t vertices to modify U to create a set W with the desired properties (removing one vertex from each "bad" set R). Using linearity of expectation one last time, we derive E[W ] ≥ E[ |U | − Z] ≥ mε t − m k0 β t , so in particular, a W with at least this size must exist, as claimed.
The below corollary simply follows by iterating the above argument, and will be convenient when we establish connections between multipartite graphs. If the edges of a graph are colored, define N c X (Y ) to be the set of common neighbors of Y in X through edges of color c. Corollary 2.5. Let r and y be positive integers, r ≥ 2. There exists N = N (r, y) with the following: let A, B 1 , . . . , B y be disjoint vertex sets, all of size n > N . Consider an r-coloring of the edges of the complete bipartite graph between A and each B i . Then there exists a set W ⊆ A, |W | = n 2 −y , and colors c 1 , . . . , c y such that for every 1 ≤ i ≤ y and every subset X ⊆ W of size 1 8 log r n we have N ci Bi (X) ≥ √ n.
Proof. We will construct a nested sequence of sets A = A 0 ⊇ A 1 ⊇ · · · ⊇ A y = W , with |A i | = n 2 −i such that every X ⊆ A i of size at least 1 8 log r n has N ci Bi (X) ≥ √ n. The definition of A 0 is clear. As an induction hypothesis, suppose that A i−1 has been defined. Let c i be the most common color in the bipartite graph between A i−1 and B i . Apply Lemma 2.4 to the graph formed by the edges of color c i , with the values m = n 2 −(i−1) , m = n, = r −1 , w = n 2 −i , k 0 = 1 8 log r n, β = 1 √ n , t = 2 −(i−1) 3 log r n, and denote by A i the resulting set W . This is possible for n large enough (say n > N i ): This proves the statement, with N = max N i .
Corollary 2.6. Let r and t be positive integers, r ≥ 2. There exists N = N (r, t) with the following: let A 1 , A 2 , . . . , A t be disjoint subsets of the vertex set of an r-colored complete graph, of size |A i | = n. Then there exist subsets X i ⊂ A i , of size |X i | = 1 2 t+1 r log r n, such that every set X i is homogeneous and every complete bipartite graph between X i and X j is monochromatic.
Proof. Induction on t. For t = 1, this follows from the multicolor version of Ramsey's theorem. Suppose that the result is true for t = i − 1, and we will prove it for t = i. Apply Corollary 2.5 with y = i − 1, A = A i and B j = A j to obtain a set W ⊂ A i and colors c 1 , . . . , c i−1 with the properties of Corollary 2.5, as long as n > N 1 . By Ramsey's theorem, W contains a homogeneous set X i ⊂ W of size 1 4r log r |W | = 1 2 i+1 log r n. By our hypothesis, for every 1 Aj (X i ) be subsets of size √ n. By the induction hypothesis, there exists N 2 such that, if √ n > N 2 , there exist subsets X j ⊂ A j such that each of them is homogeneous and the bipartite graph between every pair of them is monochromatic. Their size is |X j | = 1 2 i r log r √ n = 1 2 i+1 r log r n. This proves the statement with N = max{N 1 , N 2 2 }.
Note that for t = r we have |X i | = C r log n. We are now ready to start the proof of the main theorem.
Proof. As mentioned to previously, our goal is to find a fully-complete multipartite graph, which uses all r colors, in a large enough ε-balanced K n . In the first step of the proof, we will apply Lemma 2.3 to the r graphs induced by the edges colored in the r different color classes. The subsets we collect via the Lemma here will thus necessarily utilize all r colors. Afterwards, we will apply Corollary 2.6 to fill in for the connections between the various subsets we collected in the previous step. We now give the details. We assume n > ε −crk where c r is a sufficiently large constant that only depends on r. In the main part of the proof we assume also that k (and hence also n) is large enough with respect to r.
Given a color i ∈ [r], we apply Lemma 2.3 with parameters w = √ n, k 0 = − log ε n 8 , β = 1 √ n and t = − log ε n 3 . Thus for each color we obtain a set W i of size |W i | = √ n. These sets are not necessarily disjoint, so we take disjoint subsets W i ⊂ W i of size |W i | = √ n r . We can now apply Corollary 2.6 to find homogeneous subsets X j ⊂ W j of size |X i | = 1 2 r+1 r log r |W i | = 1 2 r+1 r 1 2 log r n − 1 ≥ − 1 2 r+3 r log ε n, pairwise joined by monochromatic graphs. Take a subset X i ⊂ X i of size |X i | = − 1 2 r+3 r log ε n.
Let U i be a subset of this common neighborhood of size √ n. Take subsets U i ⊆ U i of size √ n−r 1 2 r+3 r log ε n r ≥ 3 √ n which are pairwise disjoint and disjoint from all X i . We claim that we can take subsets U i ⊂ U i of size 4 √ n with the following property: for every v, v ∈ U i and every w ∈ ∪ r j=1 X j , the edges vw and v w have the same color.
This holds by the pigeonhole principle; one could associate a base-r vector v i of length r j=1 X j to each u i ∈ U i where v i (j) denotes the color of the edge u i sends to the j th vertex in r j=1 X j , and therefore there exists a subset |U i | with: elements that all have the same associated vector. Next we associate to each vertex w i ∈ X j a base-r vector v i of length r, v i (h) is the unique color of the edges between w i and X" h . By the pigeonhole principle, we can find a subset Y j ⊂ X j of size |X j | r r where all vertices have the same associated vector. The bipartite graph between Y j and X h is therefore monochromatic, for all j, h ∈ [r].
Finally, apply Corollary 2.6 to the sets U i to produce homogeneous sets X i of size |X i | ≥ 1 2 r+1 r log r |U i | ≥ − 1 2 r+3 r log ε n pairwise joined by monochromatic graphs. The graph induces on the vertex sets Y i and X i satisfy the properties that we want: • The sets Y i and X i are homogeneous.
• The bipartite graphs Y i Y j and X i X j are monochromatic (by Corollary 2.6), as well as the bipartite graphs Y i X j (by the pigeonhole principle).
• Every color appears in the graph, in particular the bipartite graph Y i X i has color i.

Asymmetric Patterns
In this section, we specialize on getting upper bounds on the function R ε (M l,k ).
Given subsets A and B of a 2-colored graph G, we say A is complete with B in red (blue) if all the edges in between A and B are red (blue). If A = {a}, we say a is complete with B in red (blue).
The upper bound in Proposition 1.7 follows from optimizing the below lemma for small integers.
Lemma 3.1. Let K n be an ε-balanced graph, where ε is as large as possible. Then, there exists a subset of vertices S with |S| ≥ cn such that for some x, y ∈ K n , S is complete to x in red and S is complete to y in blue.
Proof. Let K n be a ε-balanced graph, where ε is as large as possible. WLOG, K n contains at most as many red edges as blue edges. Let ∆ R and δ R denote the maximum and minimum red degrees of K n . Also, let c := √ 1 − ε − (1 − ε). Case 1: ∆ R − δ R > cn. In this case, simply consider the vertices x and y of maximum and minimum red degree, and observe that y must send at least cn blue edges into the red neighborhood of x.
Case 2: ∆ R − δ R ≤ cn. Since the average degree is εn, we must have ∆ R ≤ (ε + c)n and δ R ≥ (ε − c)n. We count the number of subgraphs of the form {x, y, z}, where {x, y} is red and {y, z} is blue. Let d R (v) denote the red degree of v. Then, the number of subgraphs of the desired form is: Then, in particular, there must be a pair of vertices with the intersection of their red and blue degree at least (ε − c)(1 − ε + c)n + o(n). Since c = (ε − c)(1 − ε + c) by choice of c, the statement of the lemma follows in this case also.
We now prove the following lemma, which is a generalization of the previous one that will allow us to prove Theorem 1.6. Proof of Lemma. Let t > 0 be a constant such that 0 < 1−t 2k < . Let 1−t 2k = α. Let r 1 , . . . , r k and b 1 , . . . , b k be the k vertices with the largest red degree and blue degree, respectively. We consider two cases: Then the set S of vertices joined by a red edge to each r i and to a blue edge to each b i has size at least tn − 2k.
There is some i such that d B (r i ) ≥ αn or d R (b i ) ≥ αn. WLOG, assume the former. Then, we have that d B (v) ≥ αn for all but at most k − 1 vertices v ∈ V (G).
Let V R be the set of vertices with d R (v) ≥ αn. We can use that G is -balanced to find a bound on |V R |: Now consider the set S of (2k+1)-tuples of distinct vertices (v, r 1 , . . . , r k , b 1 , . . . , b k ) such that, for every i, the edge vr i is red and the edge vb i is blue. If we fix v ∈ V R \{r 1 , . . . , r k }, then the number of choices of r 1 , . . . , By pigeonhole principle, there is a choice of r 1 , . . . , r k , b 1 , . . . , b k for which there are at least −α choices of v, which we can put in a set S.
Since and k do not depend on n, this proves the statement for c = min{t, −α 1−α α 2k }. Corollary 3.3. For every , r and there exists N with the following property: every -balanced K n on n > N vertices contains disjoint sets A, B, S with |A| = |B| = , |S| = k, A is complete with S in red and B is complete with S in blue. Moreover, if is fixed, then N ≤ C k(4 ) 2 .
Proof. If > 1, follow the proof above by choosing t = 1/2 and α = 1 4 . Then 0 < α < . For a value of n > C 1 k(4 ) 2 , where C 1 is large enough, we have If ≤ 1, then applying Lemma 3.2 we obtain new constant C 2 := c(ε, ε −1 ) such that for n > C 2 r there exist A , B and S with |A | = |B | = −1 , |S| = r and the desired properties. Simply take any A ⊆ A , B ⊆ B of size . This proves the statement for C = max{C 1 , C 2 }.
Observe that Theorem 1.6 follows immediately from the above Corollary replacing r with R(r).

Infinite Balanced Graphs
We now begin considering what natural restrictions we can put on our colorings to generalize our previous results to the infinite case. The main difficulty here is that it is no longer clear how the notion of being ε-balanced should generalize. For example, we cannot obtain an M ω,ω (a countably infinite analogue of an M l,r ) by merely assuming every vertex has infinite degree in both colors. Even under the stronger assumption that the back-degrees are balanced this is still not enough. The following construction applies to both restrictions. Proof. Consider the graph with vertex set consisting of two disjoint copies of N. Color all edges between vertices in the right copy blue and all edges between vertices in the left copy red. For each vertex i in the left copy and j in the right copy, color edge ij blue if i < j and red otherwise. This coloring has no M 1,ω . To satisfy the back-degree restriction, one may start with evens and odds as the partitions.
On the other hand, if we deal with the graphs defined on Polish spaces the notion of being non-meagre seems to correspond rather well with the density results from the finite version. For convenience, we recall the necessary definitions and results 1 : Let X be a topological space. If it is second countable and completely metrizable it is a Polish space, and if all of its points are limit points it is perfect. We say a set E ⊆ X is nowhere dense if its closure has empty interior. We say it is meagre if it is the union of countably many nowhere dense sets, and it is comeagre if X \ E is meagre. We say it has the Baire property ( is Baire measurable and each color class is non-meagre. Then we find some color consistent member of F r c where each blown-up clique is a Cantor set. The following results will be useful in the proof:  Recall that using the axiom of choice, one can define graphs on real numbers that fail to have a continuum sized monochromatic clique. Galvin's theorem, stated below, allows us to find a monochromatic clique which is a Cantor set (which is in particular continuum sized), if all the color classes have the BP.
Theorem 4.6 (Galvin). Suppose X is a perfect Polish space, and [X] 2 = P 0 ∪ ... ∪ P i is a partition where each P i has the BP (as subsets of X 2 ). Then there is a Cantor set C ⊆ X such that [C] 2 ⊆ P i for some i.
Proof of Theorem 4.3. Analogously to beginning with monochromatic bipartite graphs in the finite case, we start by localizing to disjoint open sets U 1 , ..., U 2r such that U 2k−1 × U 2k is comeagre in color k for k ∈ [r]. We note that this step preserves structure: for each i, j ∈ [2r] and k ∈ [r], . Therefore, if necessary, we can localize again and rename the sets to also ensure each U i × U j is comeagre in some color. Now, we begin collecting an assortment of well-behaved subsets within and between each U i . As for each i ∈ [2r], U i is open, f |[U i ] 2 is still Baire measureable. So, we can find countable intersections of dense open subsets G 1 := m∈N G m 1 , · · · , G 2r = m G m 2r where f is continuous. Further, for every i, j ∈ [2r] with i = j, U i × U j is comeagre in some color, so there is a sequence of dense open sets S m i,j for every m ∈ N such that m∈N S m i,j ⊆ U i × U j is monochromatic. We note that as each G m The discussion in the second paragraph guarantees that we can satisfy these requirements at every step. We let C i = w∈2 N n∈N R w|n i (which is well defined by the completeness of X and (2)) denote the Cantor set associated with the sequence R i . As f |[C i ] 2 must still be continuous since C i ⊆ G i by (2, 3), we may apply Galvin within each C i to obtain the desired configuration.

Discussion
Here, we we collect some open problems and future directions of research. Firstly, even though the the upper bound provided by Theorem 1.4 is asymptotically tight with respect to ε and k, the constant that arises that depends only on the number of colors c(r), was rather large. It would be interesting to obtain an upper bound of the form ε −ck , where c is sub-exponential in r (the number of colors).
We found Lemma 3.1 to be of independent interest, and we believe that the constant we obtained, √ 1 − ε−(1−ε), is not the best possible. For example, when ε = 0.5, we are able to find a subset S of size about (0.207 + o(1))n. With a slightly more elaborate counting argument, one can improve this constant to about 0.24+o (1). Paley graphs show that 0.25+o(1) is best possible. We conjecture that this is actually the truth. More generally, in the language of graphs, we conjecture the following: Conjecture 1. Let G be a graph with ε fraction of the edges. Then, there exists a subset of vertices S with |S| ≥ ε(1 − ε)n − O(1) such that there exists x, y ∈ G with S ⊆ N (x) and S ⊆ N (y) We also note that in the infinite case we considered only one of many potential ways of generalizing the notion of balanced colorings. Another rather general way is as follows: Question 1. Let ∈ (0, 0.5], X a set, F some σ-algebra on [X] 2 , and µ some probability measure on F. If we have [X] 2 = R B such that ≤ µ(R) ≤ µ(B), what structures can we guarantee appear?