On two problems in Ramsey-Turán theory

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1 On two problems in Ramsey-Turán theory József Balogh Hong Liu Maryam Sharifzadeh July 7, 016 Abstract Alon, Balogh, Keevash and Sudakov proved that the (k 1)-partite Turán graph maximizes the number of distinct r-edge-colorings with no monochromatic K k for all fixed k and r =, 3, among all n-vertex graphs. In this paper, we determine this function among n-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike the Alon-Balogh-Keevash-Sudakov s result, the extremal construction from Ramsey-Turán theory, as a natural candidate, does not maximize the number of distinct edge-coloring with no monochromatic cliques among all graphs with sub-linear independence number, even in the -colored case. In the second problem, we determine the maximum number of triangles in an n- vertex K k -free graph G with α(g) = o(n). The extremal graphs have similar structure as the extremal graphs for the classical Ramsey-Turán problem, i.e. when the number of edges are maximized. 1 Introduction Numerous classical problems in extremal graph theory have highly structured extremal configurations. For example, Turán [19] in 1941 proved that ex(n, K k ), the maximum number of edges in an n-vertex K k -free graph, is attained only by the balanced complete (k 1)-partite graph, known now as the Turán graph T n,k 1. Motivated by the fact that the Turán graph is particularly symmetric, admitting a (k 1)-partition into linear-sized independent sets, Erdős and Sós [13] introduced the Ramsey-Turán type of questions, where they investigated the maximum size of a K k -free graph G with the additional condition that α(g) = o( G ). Denote RT(n, K k, o(n)) the Ramsey-Turán function for K k, i.e. the maximum size of an n-vertex K k -free graph with independence number o(n). In 1970, Erdős and Sós [13] Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. jobal@math.uiuc.edu. Research is partially supported by NSF Grant DMS , Arnold O. Beckman Research Award (UIUC Campus Research Board 15006). Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, UK. h.liu.9@warwick.ac.uk. Supported by ERC grant and EPSRC grant EP/K01045/1. Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. sharifz@illinois.edu. 1

2 determined RT(n, K k, o(n)) for all odd k. The problem becomes much harder when an even clique is forbidden. For k = 4, Szemerédi [18], using the regularity lemma, proved that RT(n, K 4, o(n)) n /8+o(n ). It had remained an open question whether RT(n, K 4, o(n)) = Ω(n ). Bollobás and Erdős, in their seminal work [7], constructed a dense, K 4 -free graph with sub-linear independence number, matching the lower bound above (see Section for more details). For all even k, it was finally determined by Erdős, Hajnal, Sós and Szemerédi [1] in See [17] for a survey and [4, 5] for more recent development on this topic. In this paper, we will study the Ramsey-Turán extensions of some classical results, whose extremal graphs are close to the Turán graph. See e.g. [6] for one such extension of a graph tiling problem. 1.1 Edge-colorings forbidding monochromatic cliques Denote F (n, r, k) the maximum number of r-edge-colorings that an n-vertex graph can have without a monochromatic copy of K k. A trivial lower bound is given by T n,k 1 as every r-edge-coloring of a K k -free graph is monochromatic K k -free: F (n, r, k) r ex(n,k k). Erdős and Rothschild [11] in 1974 conjectured that, for sufficiently large n, the above obvious lower bound is optimal for -edge-colorings. This was verified for k = 3 by Yuster [0]. In 004, Alon, Balogh, Keevash and Sudakov [1] settled this conjecture in full, proving that, for all k 3, the Turán graph T n,k 1 maximizes the number of -edge-colorings and 3-edge-colorings with no monochromatic K k among all graphs: F (n,, k) = ex(n,k k) and F (n, 3, k) = 3 ex(n,k k). (1) For 4-edge-colorings, the only two known cases are when k = 3, 4: an asymptotic result was given in [1] for k = 3, 4; the exact result was proved by Pikhurko and Yilma [15], who showed that T n,4 and T n,9 maximizes the number of 4-edge-colorings with no monochromatic K 3 and K 4 respectively, see [16] for more recent development. Since the Turán graph is extremal in the Erdős-Rothschild problem for r =, 3, it is natural to consider its Ramsey-Turán extension. Formally, given a function f(n), we define RF(r, k, f(n)) to be the maximum number of r-edge-colorings that an n-vertex graph with independence number at most f(n) can have without a monochromatic copy of K k. Similarly, the first choice for a lower bound on RF(r, k, o(n)) is given by taking all edge-colorings on an extremal graph for Ramsey-Turán problem: RF(r, k, o(n)) r RT(n,K k,o(n))+o(n ). () Considering (1), it is not inconceivable that the lower bound in () is optimal when r is small. However, as shown in the following example, RF(r, k, o(n)) exhibits rather different behavior than F (n, r, k), even in the -edge-coloring case when K 4 is forbidden. Let G be a graph obtained by putting a copy of Γ in each partite set of T n,, where Γ is a triangle-free graph with independence number and maximum degree o(n). 1 Consider the following set of 1 The existence of K k -free graph Γ with α(γ) = o( Γ ) was proved by Erdős [9].

3 -edge-colorings of G. Color the edges inside one partite set red, the edges inside the other partite set blue, and color all the remaining cross-edges either red or blue. It is not hard to see that none of these colorings contains monochromatic K 4 s, hence, RF(, 4, o(n)) n /4, while RT(n, K 4, o(n)) = (1/8 + o(1))n. The above example already suggests an obstacle in determining RF(, k, o(n)), that is, the subgraphs induced by each color could simultaneously have linear-sized independent set. Nonetheless, our first result reveals the behavior of RF(, k, o(n)) for every integer k. Theorem 1.1. RF(, 3, o(n)) = o(n). For every integer t 1 and i {1,, 3}, RF(, 3t + i, o(n)) = RT(n,K 4t+i,o(n))+o(n ). Our constructions for the lower bound in Theorem 1.1 is based on the Bollobás-Erdős graph [7]. 1. A generalized Ramsey-Turán problem The generalized Turán-type problem, i.e. for given graphs F and H, determine ex(n, F, H), the maximum number of copies of F in an n-vertex H-free graph, has been studied for various choices of F and H. Erdős [10] determined ex(n, K s, K t ) for all t > s 3, showing that among all K t -free graphs, T n,t 1 has the maximum number of K s s. See also Bollobás and Győri [8] for ex(n, K 3, C 5 ), and more recently, Alon and Shikhelman [] for the cases when (F, H) are (K 3, C 5 ), (K m, K s,t ), and when both F and H are trees. Our second result studies the general function RT(F, H, f(n)), which is the maximum number of copies of F in an H-free n-vertex graph G with α(g) f(n). It is not hard to see that RT(K s, K s+1, o(n)) = o(n s ). We determine, in the following theorem, RT(K 3, K t, o(n)) for every integer t. Theorem 1.. Let t 6 be an integer and l = t. Then as n tends to infinity RT(K 3, K t, o(n)) = a l n 3 (1 + o(1)), where a l = max 0 x 1 ( 1 l ( l 3 ) 3 ( l 3) ) ( 1 x l ) 3 + x ( l ) ( 1 x l ) + 1 ( x ) (1 x) if t = l, if t = l + 1. (3) In fact, our proof shows that all the extremal graphs should have the structure as those in Construction.7. The definition of a l comes from optimizing the number of K 3 s in these graphs. For the general case t > s 3, we present a construction in the concluding remark which we believe gives the right answer. Our next result verifies the first non-trivial case. The case t = 5 is covered in Theorem

4 Theorem 1.3. For every s 3, RT(K s, K s+, o(n)) = ( (s ) + o(1) ) ( n s ) s. Organization. We first introduce some tools in Section. Then in Section 3, we prove Theorem 1.1, and in Section 4, Theorems 1. and 1.3. Notation. Let G = (V, E) be an n-vertex graph and e(g) = E(G). For every v V and V V, denote by d V (v) the degree of v in V. Also, let N V (v) be the set of vertices u V such that vu E(G). Denote k s (G) the number of K s in G. For every A V (G) and an r-coloring of E(G) with colors {c 1,..., c r }, let G ci [A] be the c i -colored subgraph of G induced by the vertex set A. We will write G ci instead of G ci [V (G)]. We fix throughout the paper a function ω(n) of n such that ω(n) arbitrary slowly. Preliminaries We start with a formal definition for RT(n, H, o(n)). Definition.1. For a graph H and a function f(n), let RT(n, H, o(f(n))) = lim ε 0 lim n RT(n, H, εf(n)) n. Bollobás and Erdős [7] constructed an n-vertex K 4 -free graph, BE(n), with independence number o(n) and ( o(1))n edges. We follow the description in [17] to present their construction. For a constant ε > 0, and sufficiently large integers d and n 0, let n > n 0 and µ = ε/ d. Next, partition the high-dimensional unit sphere S d into n/ domains, D 1,..., D n/, of equal measure with diameter less than µ/. For every 1 i n/, choose two points x i, y i D i. Let V (BE(n)) = X Y, where X = {x 1,..., x n/ } and Y = {y 1,..., y n/ }. For every x, x X and y, y Y, (1) let xy E(BE(n)) if their distance is less than µ, () let xx E(BE(n)) if their distance is more than µ, (3) let yy E(BE(n)) if their distance is more than µ. Note that the number of edges with both ends in X or Y is o(n ). We will also need the following definitions of regular partitions and weighted cluster graph. Definition.. Let G be a graph and A, B V (G). Denote d(a, B) := e(g[a,b]). Given A B ε > 0, a pair X, Y V (G) is ε-regular if for every A X and B Y with A ε X and B ε Y, d(a, B) d(x, Y ) ε. A vertex partition of G, V (G) = C 1... C m is ε-regular if all but εm pairs of (C i, C j ) are ε-regular. 4

5 Definition.3. For every ε > 0, positive integer t, and an n-vertex graph G = (V, E), let C = {C 1,..., C m } be an ε-regular partition of V (G) with m t. Denote R the cluster graph with respect to C and minimum density 10ε. We now define the weighted cluster graph, R = (C, w), on the vertex set C as follows. For an ε-regular pair (C i, C j ), we will define w(c i, C j ) to be the following: 0 if d(c i, C j ) 10ε or (C i, C j ) is an irregular pair, 1 if 10ε < d(c i, C j ) 1/ + 10ε, 1 if d(c i, C j ) > 1/ + 10ε. (4) Definition.4. A weighted graph G is an ordered triple (V, E, w) where E := V, set of all unordered pairs of vertices, and w : E {0, 1/, 1}. Define G 1/ = (V, E 1/ ) where E 1/ = {e E : w(e) 1/} and G 1 = (V, E 1 ) where E 1 = {e E : w(e) = 1}. Denote e(g) = e E(G) w(e). For two weighted graphs G = (V, E, w) and G = (V, E, w ), define G G = (V, E, w ) where w (e) = min{w(e), w (e)}. For X Y V, we call (X, Y ) a weighted ( X, Y )-clique or weighted complete subgraph of size l if X E 1 and Y E 1/ and X + Y = l. Also, let the weighted clique number of G be the size of the largest weighted complete subgraph of G. For a triangle T = e 1 e e 3, let w(t ) = 3 i=1 w(e i). Also, let T (G) = T T w(t ), where T is the set of all triangles in G. For X V (G), denote T (X) = T G[X] w(t ). We need the following two lemmas from [1], the first one has been proved in the proof of Theorem in [1]. Lemma.5. For every n-vertex graph G, if its weighted cluster graph R(C, w) contains a weighted clique (X, Y ) of size l such that α(g[x]) = o(n), then G contains a copy of K l. Lemma.6. For every integer k 3 and an n-vertex weighted graph G = (V, E, w), if G does not contain a weighted complete subgraph of size k, then e(g) (b k + o(1))n, where It was proved in [1] that b k = 1. k 3 k k 10 3k 4 if k is odd, if k is even. (5) RT(n, K k, o(n)) = (b k + o(1))n, (6) where the definition of b k comes from optimizing the number of edges in the construction below. Additionally, it was proved that some of the extremal graphs are from the following construction. 5

6 Construction.7. Given k 3, let l = k. Let Γ n be an n-vertex triangle-free graph with α(γ n ) = o( Γ n ). Denote H(n, k) the family of n-vertex graphs G obtained as follows. Start with a complete l-partite graph on vertex set V 1... V l. If k is odd, then put a copy of Γ Vi in each V i. If k is even, for each i 3, put a copy of Γ Vi in each V i, and replace G[V 1 V ] by a Bollobás-Erdős graph, BE( V 1 V ), keeping naturally the classes. We will use the following multicolored version of the Szemerédi regularity lemma (for example, see [14]). Theorem.8. For every ε > 0 and integer r, there exists an M such that for every n > M and every r-coloring of the edges of an n-vertex graph G with colors {c 1,..., c r }, there exists a partition of V (G) into sets V 1,..., V m with V i V j 1, for some 1/ε < m < M, which is ε-regular with respect to G ci for every 1 i r. 3 Proof of Theorem 1.1 To overcome the obstacle that all subgraphs induced by each color could have linear-sized independent set, we need the following simple, but somewhat surprising observation. Lemma 3.1. For every c > 0 and r, set a = a r (c) = c 3 r 1. Let G be an n- vertex graph with α(g) a r n and C : E(G) {c 1,..., c r }. Then there exists a partition V (G) = C 1... C r such that α(g ci [C i ]) cn for every 1 i r. Proof. We will use induction on the number of colors. For the base case when r =, we may assume α(g c1 ) > cn and α(g c ) > cn. Let X 0 = V (G), Y 0 = and a = c. We iterate the following operation. At step i, if α(g c1 [X i 1 ]) cn then we will stop. Otherwise, let I be a maximum independent set in G c1 [X i 1 ]. Since α(g) an, we have α(g c [I]) an. We define X i := X i 1 \ I and Y i := Y i 1 I. Notice that α(g c [Y i ]) α(g c [Y i 1 ]) + an. Suppose the iteration stops after k steps, i.e. α(g c1 [X k ]) cn, then k n = 1/c, which cn implies that α(g c [Y k ]) k an cn as desired. For the inductive step, let us assume that the lemma holds for r 1 colors. In particular, for every a > 0, n -vertex graph H with α(h) an, and (r 1)-edge-coloring of H with colors c 1,..., c r 1, there exists a partition of V (H) = C 1... C r 1 such that α(h c i [C i]) a 1/(3 r 3 1) n for all 1 i r 1. Now, we will prove the lemma for r colors. Fix an arbitrary r-edge-coloring of G, we can assume that α(g c1 ) > cn. Let C 1,0 = V (G) and C i,0 = for all i r. We iterate the following operation. At step k, if α(g c1 [C 1,k 1 ]) cn then we will stop. Otherwise, let I be a maximum independent set of G c1 [C 1,k 1 ] with n > cn vertices. Since α(g) a r n, we have α(g c [I]... G cr [I]) a r n < a r n /c. We can apply the induction hypothesis to the graph G[I]. Therefore, there exists a partition of I = I... I r such that for every i r, we have α(g ci [I i ]) ( ar ) 1 3 r 3 1 c n = c 3 r 3 r 3 1 n c 3 r 3 r 3 1 n. (7) 6

7 Then, we define C 1,k := C 1,k 1 \ I and C i,k := C i,k 1 I i, for i r. By (7), α(g ci [C i,k ]) α(g ci [C i,k 1 ]) + c 3 r 3 r 3 1 n. Let us assume that the iteration stops after l steps, i.e. α(g c1 [C 1,l ]) cn. Note that l = 1/c, which implies that for every i r, n cn α(g ci [C i,l ]) l c 3 r 1 3 r 3 1 n c c 3 r 3 r 3 3 r 1 n = c 3 r n = cn. Proof of Theorem 1.1. (Lower bound) Fix an arbitrary t 1. For each i = 1,, 3, by (6), there is a graph G i in H(n, 4t + i) from Construction.7 with e(g i ) = RT(n, 4t + i, o(n)) = (b 4t+i + o(1))n. We take the following set of colorings. i = 1: Set V (G 1 ) = V 1... V t. Color G 1 [V p ], for 1 p t, red, color G 1 [V q ], for t + 1 q t, blue, and all cross-edges in G 1 [V p, V q ], 1 p < q t, in either red or blue. i = : Set V (G ) = V 1... V t+1. Color G [V p ], for 1 p t + 1, red, color G [V q ], for t + q t + 1, blue, and all cross-edges in G [V p, V q ], 1 p < q t + 1, in either red or blue. i = 3: Set V (G 3 ) = V 1... V t+1. Color G 3 [V p ], for 1 p t, red, color G 3 [V q ], for t + 1 q t + 1, blue, and all cross-edges in G 3 [V p, V q ], 1 p < q t + 1, in either red or blue. It is easy to check, for i = 1, 3, that in each G i, we obtain RT(n,4t+i,o(n))+o(n) -edgecolorings without monochromatic K 3t+i. To see the case when i =, fix arbitrary p, q such that 1 p t + 1 and t + q t + 1. Note that to get a blue clique, we can have at most 1 vertex from each V p and vertices from each V q, hence, the largest blue clique has size at most 1 (t + 1) + t = 3t + 1. For a red clique, we can have at most 1 vertex from each V q, at most a K 3 from V 1 V, and at most vertices from each V p, p 1,. Thus, the largest red clique is of size at most 3 + (t 1) + 1 t = 3t + 1. (Upper bound) For an arbitrary small constant ε > 0, let G be a graph with α(g) = ε n. For any fixed -edge-coloring of G, φ : E(G) {φ 1, φ }, apply Lemma 3.1 with r =, a = ε, and c = ε. Let {A, B} be the resulted partition such that α(g φ1 [A]) εn, and α(g φ [B]) εn. (8) We then apply Theorem.8 to G with coloring φ, and let P = {P 1,..., P m } be the resulted partition of V (G). Note that we may assume the regularity partition P refines the 7

8 {A, B}-partition. Let R φ1 and R φ be the φ 1 -colored and φ -colored weighted cluster graphs respectively. We have number of ways to fix an {A, B}-partition of V (G) n, number of ways to fix a P-partition of V (G) m n ( number of ways to fix R φ1 and R φ (m ) ) 4. Now, we will count the number of colorings with a fixed {A, B}-partition, P-partition and weighted cluster graphs R φ1 and R φ. First, note that the number of edges with both ends in one of the P i s, in between irregular pairs or sparse pairs is at most o(n ). Hence, the number of ways to color these edges is at most o(n), which implies RF(, 3t + i, o(n)) ( n m) e(r φ1 R φ )+o(n ). To complete the proof, it remains to show that (i) when φ is monochromatic K 3 -free, e(r φ1 R φ ) = 0, and (ii) when φ is monochromatic K 3t+i -free for t 1 and i = 1,, 3, e(r φ1 R φ ) RT (m, 4t + i, o(m)) + o(m ) = b 4t+i m + o(m ), (9) where b 4t+i is defined in (5). To see (i), notice that if there is an edge, say uv E(R φ1 R φ ), then, without loss of generality, we may assume that u A. Therefore, by setting X = {u} and Y = {u, v}, it follows from (8) that we have a φ 1 -colored weighted (1, )-clique (X, Y ) with G φ1 [X] = o(n), which by Lemma.5 yields a monochromatic K 3 in φ, a contradiction. For (ii), suppose that (9) is not satisfied, then by Lemma.6, the graph R φ1 R φ has a weighted complete subgraph (X, Y ) of size 4t + i, and we shall find a monochromatic K 3t+i in G using (X, Y ), which is a contradiction. Let x = X, y = Y and X = {P 1... P x }. Without loss of generality, we may assume that P 1... P x/ := X A, i.e. α(g φ1 [P 1... P x/ ]) = o(n). We have thus find a weighted clique (X, Y ) in G φ1 such that the independence number of G φ1 [X ] is o(n) = o( G[X ] ). Hence, Lemma.5 shows that G φ1 contains a copy of K x/ +y. Claim 3.. x t. Proof. Suppose that x t + 1. Recall that from the definition of weighted clique that X Y, i.e. x y, and x + y = 4t + i. Thus, x 4t + 3 4t + i = x + y x 4 4(t + 1), a contradiction. Claim 3. then implies that monochromatic clique we found in G φ1 is of order x x + y = x + y 4t + i t = 3t + i, (10) a contradiction. 8

9 4 Proof of Theorems 1. and 1.3 Proof of Theorem 1.3. (Lower bound) Let G be an n-vertex graph with a balanced vertex partition V 1,..., V s. The vertex sets V 1,..., V s are the same set of uniformly distributed points from the high dimensional sphere, as in the Bollobás-Erdős graph. The edge set of G[V i V j ] is the same copy of BE(n/s), for all 1 i, j s. Note that each G[V i ] is triangle-free and each G[V i V j ] is K 4 -free. We claim that G is K s+ -free. Indeed, let F be a largest clique in G and let g i = V (F ) V i. Since G[V i ] is triangle-free, each g i. If V (F ) s +, then there exists at least two indices p, q such that g p = g q =, which contradicts to G[V p, V q ] being K 4 -free. We will count the number of K s with exactly one vertex from each V i. Fix a vertex v 1 V 1, a uniformly at random chosen v V is adjacent to v 1 if v is in the cap (almost a hemisphere) centered at v 1 with measure 1/ o(1), which happens with probability 1/ o(1). Now we fix a clique on vertex set {v 1,..., v l 1 } with l and v i V i. The number of vertices in V l that are in l 1 i=1 N(v i) is at least (l 1) n/s o(n). Therefore, we have k s (G) s i=1 [ ( (i 1) o(1) ) n ] ( ) ( = (s ) n ) s o(1). s s (Upper bound) Let G be an n-vertex K s+ -free graph with α(g) = o(n). Let R be the weighted cluster graph of G. We call an edge in R heavy if it has weight 1. By Lemma.5, R does not contain any weighted (1, s + 1)-, or (, s)-clique. In other words, we have that R is K s+1 -free and does not have a copy of K s with at least one heavy edge. Since the number of K s in the K s+1 -free graph R is at most ( R /s) s and not having a K s with a heavy edge implies that each K s in R has weight at most (s ) + o(1). Hence, since the number of Ks not spanning s different clusters in R is negligible, i.e. o(n s ), we have that the number of K s in G is at most ( R s ) s (s ) ( ) s G ( ) ( + o(n s ) = (s ) n ) s + o(1). R s The following lemma is the main step for proving Theorem 1.. Lemma 4.1. For every integer t 4 and n-vertex weighted graph G = (V, E, w) (as in Definition.4) with no weighted complete subgraph of size t, we have where a t is as in (3). T (G) a t n 3 (1 + o(1)), (11) Proof. Let G = (V, E, w) be an n-vertex weighted graph that satisfies the hypothesis and has the maximum number of triangles. First, we will apply two rounds of the so-called symmetrization method to the graph G. For v, v V (G), denote T v (G) = v T G w(t ), the 9

10 number of weighted triangles containing v. Similarly define T vv (G) = v,v T G w(t ). Let V (G) = {v 1,..., v n } such that T v1 (G)... T vn (G). Define S 1 (i, j), i j [n], to be the following operation: if v i v j / G 1/ then we replace v j with a copy of v i, i.e. change w(v j v k ) to w(v i, v k ) for all k i, j. If i < j, then the number of triangles changes by T vi (G) T vj (G) 0. Since G is maximal, we have that T vi (G) = T vj (G) for any v i v j / G 1/. Consequently, the following process, denoted by S 1, is finite: apply S 1 (i, j) for every 1 i < j n with v i v j / G 1/. Note that S 1 will not increase the weighted clique number and keep the same number of triangles. After S 1, in the resulting graph v i v j / G 1/ is an equivalence relation. Denote A = {A 1,..., A m } the equivalence classes of this relation, i.e. two vertices u and v are in the same class if and only if uv / G 1/. Therefore, for fixed 1 i, j m, all the edges between A i and A j have equal weights, which we denote by w(a i A j ), and for all vertices x, x A i and y, y A j, we have T x (G) = T x (G) and T xy (G) = T x y (G). Therefore, we can define T Ai (G) = T x (G) and T Ai A j (G) = T xy (G). Note that if (X, Y ) is one of the largest weighted complete subgraphs of G, then Y = m. We summarize the structure of G as follows: Let H be a weighted complete graph on vertex set {a 1,..., a m } with all its edges having weight either 1 or 1/, and w(a i a j ) = w(a i A j ). The graph G is a blow-up of H where we replace each a i with a set of A i vertices, and inside each A i the weight of all edges is zero. Our next goal is to show that a second round of symmetrization can be carried out in G, in other words, in H, w(a i a j ) = 1/ is an equivalence relation. Without loss of generality we may assume T A1 (G)... T Am (G). For every 1 i, j m, define S (i, j) to be the following operation: Change w(a j A k ) to w(a i A k ) for all k i, j, and denote G Ai the resulting graph. Define G Aj analogously as the graph obtained from applying S (j, i) to G. The following claim states that when w(a i A j ) = 1/, we can replace vertices in A i with copies of vertices in A j, or the other way around, without decreasing the number of triangles. Claim 4.. For every pair of integers 1 i < j m with w(a i A j ) = 1/, (i) T Ai (G) = T Aj (G); (ii) k 3 (G Ai ) = k 3 (G Aj ) = k 3 (G). Proof. Define T o A i (G) = T Ai (G) T Ai A j (G) and G = G \ {A i A j }. Since T Ai (G) T Aj (G), we have T o A i (G) + T Ai A j (G) T o A j (G) + T Ai A j (G) T o A i (G) T o A j (G). (1) For (i), it suffices to show T o A i (G) = T o A j (G). (13) 10

11 Note that k 3 (G) = k 3 (G ) + A i T o A i (G) + A j T o A j (G) + A i A j T Ai A j (G), (14) k 3 (G Ai ) = k 3 (G ) + ( A i + A j ) T o A i (G) + A i A j T Ai A j (G Ai ). (15) Then, since G is maximal, 0 k 3 (G Ai ) k 3 (G) = A j (T o A i (G) T o A j (G)) + A i A j (T Ai A j (G Ai ) T Ai A j (G)). Therefore, by (1), we only need to show T Ai A j (G Ai ) T Ai A j (G). Let { V 1,1/ = A l : w(a i A l ) = 1 and w(a j A l ) = 1 }, { V 1/,1 = A l : w(a i A l ) = 1 } and w(a ja l ) = 1, { V 1/,1/ = A l : w(a i A l ) = 1 and w(a ja l ) = 1 }, V 1,1 = {A l : w(a i A l ) = 1 and w(a j A l ) = 1}. Denote V p,q = A l V p,q A l for p, q {1/, 1}. We have T Ai A j (G Ai ) T Ai A j (G) = ( 1 1 ) ( 1 V 1,1/ ) V 1/,1 = V 1,1/ 1 8 V 1/,1. Therefore, it suffices to show V 1,1/ V 1/,1. For the sake of contradiction, assume V 1/,1 > V 1,1/. (16) We will show that (16) contradicts the maximality of G. Note that k 3 (G Aj ) = k 3 (G ) + ( A i + A j ) T o A j (G) + A i A j T Ai A j (G Aj ). (17) By (14), (15), (17), and the maximality of G we have ( 1 k 3 (G Ai ) k 3 (G) 4 V 1,1/ 1 ) 8 V 1/,1 A i A j + A j TA o i (G) A j TA o j (G), (18) ( 1 k 3 (G Aj ) k 3 (G) 4 V 1/,1 1 ) 8 V 1,1/ A i A j + A i TA o j (G) A i TA o i (G). (19) Then (18) and (19) imply ( 1 4 V 1,1/ 1 ) 8 V 1/,1 A i + TA o i (G) TA o j (G), ( 1 4 V 1/,1 1 ) 8 V 1,1/ A j + TA o j (G) TA o i (G). 11

12 Therefore, ( 1 4 V 1/,1 1 ) ( 1 8 V 1,1/ A j TA o i (G) TA o j (G) 8 V 1/,1 1 ) 4 V 1,1/ A i 1 8 V 1/,1 ( A j A i ) 1 8 V 1,1/ ( A j A i ) (16) < 1 16 V 1/,1 ( A j A i ) 4 A j A i < A j A i 4 A j < A j, a contradiction. For (ii), by the maximality of G, it suffices to show that k 3 (G Ai ) + k 3 (G Aj ) k 3 (G). By (13), (14), (15) and (17), we have k 3 (G Ai ) + k 3 (G Aj ) k 3 (G) = A i A j (T Ai A j (G Ai ) + T Ai A j (G Aj ) T Ai A j (G)). It is left to show that T Ai A j (G Ai ) + T Ai A j (G Aj ) T Ai A j (G) 0. Indeed, T Ai A j (G Ai ) + T Ai A j (G Aj ) T Ai A j (G) = 1 w(a i A k ) A k + 1 = 1 1 k m, k i,j 1 k m, k i,j 1 k m, k i,j (w(a i A k ) w(a j A k )) A k 0. w(a j A k ) A k 1 1 k m, k i,j w(a i A k )w(a j A k ) A k Denoted by S the following process: let σ be the lexicographical ordering of ( ) [m] and apply S (i, j), according to σ, for all pairs (i, j) with w(a i A j ) = 1/. By Claim 4., S is finite and keeps the number of triangles. Claim 4.3. The operation S does not change the weighted clique number of G. Proof. Let (X, Y ) be one of the largest weighted complete subgraph of G of size l. Note that Y is still m. Also, since we only repeat this operation for vertices x and y with w(xy) = 1/, the operation is not changing X either. Hence, after repeated applications of this operation, the weighted clique number of G will not change. After applying S, we have an equivalence relation on classes A 1,..., A m, which naturally extends to V (G). To be precise, denote B = {B 1,..., B m } the equivalence classes of this relation, i.e. two vertices u and v are in the same class if and only if uv / G 1. Then, the A-partition is a refinement of the B-partition. More importantly, the size of the largest weighted complete subgraph is m + m. We will next show that we can perform some transformations (Claims 4.4 and 4.5) to get a more structured graph (as those in Construction.7) without increasing the weighted clique number and decreasing the number of triangles. 1

13 Claim 4.4. Each B i contains at most two A j s. Proof. Let us assume that B 1 contains k A j s, A 1,..., A k, where k 3. Denote U the vertex set of B 1 and write u = U. Note that the edges between two B i s always have weight 1 and the edges inside an A i have weight 0 and all the other edges have weight 1/. We will divide the proof into three cases depending on the value of k. In each case, we will modify B 1 by splitting it into multiple parts. This modification will only change the weight of the edges with both ends in U and also the equivalence classes A and B. Then we need to prove that the weighted clique number did not increase, and the number of triangles increases. For the latter, since the weight of the edges with at least one end in V \ U remain the same, we only need to show that the number of triangles with two or three vertices in U did not decrease. Therefore, it suffices to show that both e(u) and T (U) increase. Case 1: Assume k 5, which implies u 5. We will split vertices in U into three parts, B 11, B 1 and B 13, such that B 11 B 1 B 13 B Also, define A i = B 1i for all 1 i 3. For every u U and v V \U, we will not change w(uv). For all vertices u, u U if they belong to the same B 1i, let w(uu ) = 0, otherwise let w(uu ) = 1. The equivalence classes A and B will change to {A 1, A, A 3, A k+1,..., A m } and {B 11, B 1, B 13, B,..., B m }. Since k 5, the number of classes in the A partition decreased by at least two and the number of classes in the B partition increased by exactly, hence, the weighted clique number of G will not increase. Now, we only need to show that the number of triangles in the graph G increases. ( ) k u before: e(u) k 1 < u 4, 3 u = u if u 0 (mod 3), 9 3 (u 1) after: e(u) = + (u 1)(u+) = 9 9 u 1 if u 1 (mod 3), 3 (u+1) + (u )(u+1) = u 1 if u (mod 3) Therefore e(u) increases for u. Now, for T (U) we have ( ) k u 3 before: T (U) 3 k u3 48, u 3 if u 0 (mod 3), 7 (u 1)(u 1)(u+) after: T (U) = if u 1 (mod 3), 7 (u )(u+1)(u+1) if u (mod 3), 7 which means that T (U) increases if u 3. Case : Assume k = 4 which implies u 4. Let us split vertices in U into three parts A 1, A and A 3, such that A 1 A A 3 A Also let B 11 = A 1 A and B 1 = A 3. For all vertices u, u U if they are in different B 1i s then w(uu ) = 1. If they are both in B 11 but in different A i s then w(uu ) = 1/, and w(uu ) = 0 if they are in the same A i. The equivalence classes A and B will change to {A 1, A, A 3, A 5,..., A m } and {B 11, B 1, B,..., B m }. Notice that the number of classes in the A partition decreased by 13

14 one and the number of classes in B increased by one, hence, the weighted clique number of G will not change. For e(u): ( ) 4 u before: e(u) 16 1 = 3u 16, 1 u + u if u 0 (mod 3), after: e(u) = (u 1)(u 1) + (u 1)(u+) if u 1 (mod 3), (u )(u+1) + (u )(u+1) + (u+1)(u+1) if u (mod 3) For u, e(u) increases. We also need to show that T (U) increases: before: T (U) 4 u = u , 1 u3 if u 0 (mod 3), 7 1 after: T (U) = (u 1)(u 1)(u+) if u 1 (mod 3), 7 1 (u )(u+1)(u+1) if u (mod 3). 7 Therefore T (U) will increase for u 3. Case 3: Assume k = 3, which implies u 3. First, suppose that u 1n/13. Split vertices in U into two equal parts B 11 and B 1. Also define A 1 = B 11 and A = B 1. For all vertices u, u U if they are in different B 1i s then set w(uu ) = 1, and w(uu ) = 0 otherwise. The equivalence classes A and B will change to {A 1, A, A 4,..., A m } and {B 11, B 1, B,..., B m }. Notice that the number of classes in the A partition decreased by one and the number of classes in the B partition increased by one, hence, the weighted clique number of G will not change. For the change on the number of triangles, we have before: T (U) + e(u)(n u) u ( ) 3 u 3 (n u), 9 { 0 + u (n u) if u is even, after: T (U) + e(u)(n u) = (u 1)(u+1) (n u) u (n u) if u is odd Since u 1n/13, we have u u u (n u) (n u) u 1 (n u) u 1 13 n. u3 u (n u) We may now assume that u > 1n/13. Let U be the vertex set of B and u = B. Since u 1n/13 and u n/13, we may assume B contains at most two A i s. Note that u u/1. We split U U into three classes of the same size, B 0, B 1 and B. Define A 0 = B 0, A 1 = B 1, and A = B. For two vertices u, u U U, if they belong to the same B i then w(uu ) = 0, otherwise w(uu ) = 1. The equivalence classes A and B will change to {A 0, A 1, A, A 5,..., A m } and {B 0, B 1, B, B 3,..., B m }. Notice that the number of classes 14

15 in A decreased by one and the number of classes in B increased by one, which implies that the weighted clique number of G will not change. We are left to show that this operation will increase e(u U ) and T (U U ): before: e(u U ) u uu + u 8 u 6 + u 1 + u 8 = 3u 1 + u 8, 3 (u+u ) if u + u 0 (mod 3), after: e(u U 9 (u+u ) = 1)(u+u 1) + (u+u 1)(u+u +) if u + u (mod 3), (u+u )(u+u +1) + (u+u +1) if u + u (mod 3). 9 9 Hence e(u U ) is increasing for u 3. We also have ( u ) 3 before: T (U U 1 ) u 3 u + u u 8 u3 6 + u u3 8 1 u3 51.5, (u+u ) 3 if u + u 0 (mod 3), after: T (U U 7 (u+u ) = 1)(u+u 1)(u+u +) if u + u 7 1 (mod 3), (u+u )(u+u +1)(u+u +1) if u + u (mod 3). 7 Therefore T (U U ) increases for u + u 3. Claim 4.5. There is at most one B i that contains two A j s. Proof. Now, we know that no B i contains three or more A i s. Let us assume that B 1 = A 1 A and B = A 3 A 4. Denote U the vertex set of B 1 B, and write u = U 4. We will split the vertices in U into three equal pieces, B 11, B 1 and B 13, and define A 1 = B 11, A = B 1, and A 3 = B 13. For two vertices u, u U if they are in two different B 1i s then w(uu ) = 1, otherwise w(uu ) = 0. This operation will change A and B to {A 1, A, A 3, A 5,..., A m } and {B 11, B 1, B 13, B 3,..., B m }, therefore the weighted clique number does not change. We only need to show that e(u) and T (U) increase. before: e(u) u 4 + u 4 = 5u 16, 3 u if u 0 (mod 3), 9 (u 1) after: e(u) = + (u 1)(u+) if u 1 (mod 3), 9 9 (u+1) + (u )(u+1) if u (mod 3), 9 9 before: T (U) u 16 u = u3 3, u 3 if u 0 (mod 3), 7 (u 1) after: T (U) = (u+) if u 1 (mod 3), 7 (u+1) (u ) if u (mod 3). 7 It can be easily checked that for u 4, both e(u) and T (U) are not decreasing. 15

16 Now, we will use the Claims 4.4 and 4.5 to complete the proof of Lemma 4.1. Let us assume that the extremal graph has partitions A = {A 1,..., A m } and B = {B 1,..., B m }. Also, since A is a refinement of B and also by Claims 4.4 and 4.5, we have m m m + 1. For the case t = l + 1, the graph does not contain a weighted clique of size l + 1, which implies m + m l. Therefore m = m = l will maximize the number of triangles. In particular, the extremal graph is an l-partite graph with partite sets B 1... B l, where B i B j 1 for all 1 i < j l. Define A i = B i for all 1 i l, and for two vertices u and v if they belong to two different B i s then w(uv) = 1, otherwise w(uv) = 0. For the case t = l, the graph does not contain a weighted clique of size l which implies m + m l 1. Therefore, in the extremal example, m = l 1 and m = l. Hence, the extremal example is an (l 1)-partite graph, with partite sets B 1... B l 1, where B 1 = x and B i = (n x)/(l ) for all i l 1. Also let A 1 A = B 1 such that A 1 A 1, and A i+1 = B i for all i l 1. For two vertices u and v if they belong to two different B i s then w(uv) = 1. Otherwise, if they both belong to B 1 but to different A i s then w(uu ) = 1/, and w(uu ) = 0 in all other cases. Now, we only need to maximize the number of triangles with respect to x, which completes the proof of Lemma 4.1. Proof of Theorem 1.. The lower bound comes from H(n, k) with k = t in Construction.7. In this case, we solve an optimization problem to find the size of V i s that maximizes the number of triangles, which is how a l is defined in (3). For the upper bound, let G be an n-vertex K t -free graph with α(g) = o(n). Let R(C, w) be the weighted cluster graph of G. By Lemma.5, we have that R(C, w) does not contain a weighted clique of size t. Then the upper bound follows from Lemma 4.1 and k 3 (G)/ G 3 = (1 + o(1))t (R)/ R 3. 5 Concluding remarks In this paper, we study the Ramsey-Turán extensions of classical problems. We determine RF(, k, o(n)), that is, the maximum number of -edge-colorings an n-vertex graph with independence number o(n) can have without a monochromatic K k, and RT(K 3, K t, o(n)), the maximum number of triangles in an n-vertex K t -free graph with o(n) independence number edge-colorings The Ramsey-Turán extension of the Erdős-Rothschild problem for more than colors remains widely open. It is known [1] that F (n, 3, k) = 3 ex(n,k k). It will be interesting to study for 3-edge-colorings, RF(3, k, o(n)). The following determines the case when forbidding monochromatic triangles. We give here only a sketch of a proof. Theorem 5.1. RF(3, 3, o(n)) = n /4+o(n ). 16

17 Sketch of a proof. (Lower bound) Let G H(n, 5) with V 1 = V = n/. Consider the following 3-edge-colorings. Color edges in G[V i ], i = 1,, red and color the cross-edges G[V 1, V ] either green or blue. (Upper bound) Let G be an extremal graph, φ : E(G) {φ 1, φ, φ 3 } be a 3-edge-coloring with no monochromatic K 3, and (A 1, A, A 3 ) be the partition obtained from Lemma 3.1 such that α(g φi [A i ]) = o(n). Let R be the multigraph by taking the union 3 i=1r φi, where R φi is the cluster graph in color φ i. Denote by µ i, i = 1,, 3, the edge-density of the subgraph of R induced by edges with multiplicity i. Note first that µ 3 = 0, since otherwise a multiplicity- 3 edge results in a weighted clique of size 3 in i R φi, contradicting to φ containing no monochromatic K 3. It suffices then to show µ 1/. Notice that no φ i -colored edge can have an endpoint in A i, otherwise we have a φ i -colored triangle. This implies that (i) for every i j [3], all edges in R [A i, A j ] have multiplicity 1 with color φ k, k i, j; (ii) for i [3], all edges in R [A i ] have multiplicity at most, colored in {j, k} = [3] \ {i}. By (i), we only need to consider edges in i R [A i ]. By (ii), inside A i, there is no color φ i. This together with the observation that edges colored in {φ p, φ q } for any p q [3] is triangle-free, we have µ 1/ as desired. 5. Generalized Ramsey-Turán for larger cliques It seems plausible that for the general case RT(K s, K t, o(n)), t > s 3, some graph from the following construction has the maximum number of K s. Construction 5.. Given 3 s < t s 1, denote by H(n, s, t) the family of n-vertex graphs G on vertex set V 1... V s obtained as follows. Let H be an extremal K t s -free graph on vertex set [s]. Make [V i, V j ] complete bipartite if ij E(H); otherwise, put a copy of BE( V i V j ) if ij E(H). For every i V (H) with d H (i) = s 1, put a V i -vertex triangle-free graph with o( V i ) independence number in V i. Note that all graphs G in the above construction are K t -free and have o(n) independence number. Indeed, since G[V i ] is triangle-free for all i, in order to have a copy of K t, there should be at least t s classes, V j1,..., V jt s, each containing vertices that form a K (t s). This would imply for every 1 p < q t s, G[V jp, V jq ] contains a K 4, which contradicts to H being K t s -free. It should also be noted that the sizes of V i s need to be optimized. Conjecture 5.3. Given integers t > s 3, one of the extremal graphs for RT(K s, K t, o(n)) lies in H(n, s, t) from Construction 5. when t s 1, and lies in H(n, k) with k = t from Construction.7 when t s. 5.3 Phase transition For a given graph H and two function f(n) g(n), we say that the Ramsey-Turán function for H exhibits a jump or has a phase transition from g(n) to f(n) if lim sup n RT(n, H, f(n)) n < lim inf n 17 RT(n, H, g(n)) n.

18 Let g r (n) = n ω(n) log1 1/r n. Balogh, Hu and Simonovits [3] showed that the Ramsey- Turán function for the even clique K r exhibits a jump from o(n) to g r (n). A similar phenomenon happens in the more general setup. Theorem 5.4. (i) RT(K 3, K 5, g 3 (n)) = o(n 3 ) and RT(K 3, K 6, g 3 (n)) = o(n 3 ). (ii) Odd cliques larger than 5 are stable: for every l 3, RT(K 3, K l+1, g l+1 (n)) = (1 + o(1))rt(k 3, K l+1, o(n)). (iii) Even cliques always exhibit a jump: for every l 3, RT(K 3, K l+, g l+1 (n)) = (1 + o(1))rt(k 3, K l+1, o(n)). We will need a lemma by Balogh-Hu-Simonovits (Claim 6.1 in [3]). Lemma 5.5. Let G be an n-vertex graph with α(g) = g q (n), where g q (n) = n w(n) log1 1/q n and ω(n) arbitrary slowly. If there exists a K q in the cluster graph of G, then K q G. Proof of Theorem 5.4. For (i), note that RT(K 3, K 5, g 3 (n)) RT(K 3, K 6, g 3 (n)). Therefore, we only need to prove RT(K 3, K 6, g 3 (n)) = o(n 3 ). By Lemma 5.5, if G is an n-vertex K 6 - free graph with α(g) g 3 (n) then the cluster graph of G is K 3 -free, which means that k 3 (G) = o(n 3 ). For (ii), note that RT(K 3, K l+1, g l+1 (n)) RT(K 3, K l+1, o(n)), hence, by Theorem 1., it is sufficient to prove RT(K 3, K l+1, g l+1 (n)) (1 + o(1)) ( ( l n ) 3. 3) l Construction.7 shows that this inequality holds. For (iii), note that RT(K 3, K l+1, g l+1 (n)) RT(K 3, K l+, g l+1 ). Hence, using (ii), we only need to show that RT(K 3, K l+, g l+1 (n)) RT(K 3, K l+1, o(n)). By Lemma 5.5, if G is an n-vertex K l+ -free graph with α(g) g l+1 (n) then the cluster graph of G is K l+1 -free. Then, by the result of Erdős [10], among all K l+1 -free graphs the l-partite Turán graph has the maximum number of triangles. Hence, we have ( ) l (n ) 3 RT(K 3, K l+, g l+1 (n)) (1 + o(1)) = RT (K3, K l+1, o(n)), 3 l where the last equality is by Theorem 1.. References [1] N. Alon, J. Balogh, P. Keevash, B. Sudakov, The number of edge-colorings with no monochromatic cliques, J. London Math. Soc. (), 70, (004), [] N. Alon, C. Shikhelman, Many T copies in H-free graphs, J. Combin. Theory Ser. B, to appear. 18

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