Asymptotics of Ramsey numbers of double stars
AASYMPTOTICS OF RAMSEY NUMBERS OF DOUBLE STARS
SERGEY NORIN, YUE RU SUN, AND YI ZHAO
Abstract. A double star S ( n, m ) is the graph obtained by joining the center of a star with n leaves to a center of a star with m leaves by an edge. Let r ( S ( n, m )) denote the Ramseynumber of the double star S ( n, m ). In 1979 Grossman, Harary and Klawe have shown that r ( S ( n, m )) = max { n + 2 m + 2 , n + 2 } for 3 ≤ m ≤ n ≤ √ m and 3 m ≤ n . They conjectured that equality holds for all m, n ≥
3. Using a flag algebra computation, we extend their result showing that r ( S ( n, m )) ≤ n + 2 m + 2 for m ≤ n ≤ . m . On the other hand, we show that the conjecture failsfor m + o ( m ) ≤ n ≤ m − o ( m ). Our examples additionally give a negative answer toa question of Erd˝os, Faudree, Rousseau and Schelp from 1982. Introduction
The Ramsey number r ( G ) of a graph G is the least integer N such that any 2-coloringof edges of K N contains a monochromatic copy of G . The difficult problem of estimatingRamsey numbers of various graph families has attracted considerable attention since itsintroduction in the paper of Erd˝os and Szekeres [7]. See [4, 14] for recent surveys. ComputingRamsey numbers exactly appears to be very difficult in general, even for trees. However,determining the Ramsey numbers of stars is fairly straightforward. Harary [9] has shownthat r ( K ,n ) = (cid:40) n, if n is odd , n − , if n is even . A natural direction in extending the above result is to consider double stars. A double star S ( n, m ), where n ≥ m ≥
0, is the graph consisting of the union of two stars, K ,n and K ,m ,and an edge called the bridge , joining the centers of these two stars. Grossman, Harary andKlawe have established the following bounds on r ( S ( n, m )). Theorem 1.1 (Grossman, Harary and Klawe [8]) . r ( S ( n, m )) = (cid:40) max(2 n + 1 , n + 2 m + 2) if n is odd and m ≤ , max(2 n + 2 , n + 2 m + 2) if n is even or m ≥ , and n ≤ √ m or n ≥ m, They further conjectured that the restriction n ≤ √ m or n ≥ m is not necessary. Conjecture 1.2 (Grossman, Harary and Klawe [8]) . r ( S ( n, m )) ≤ max(2 n + 2 , n + 2 m + 2)for all n ≥ m ≥ m and n . The first two authors are supported by an NSERC grant 418520. The third author is partially supportedby NSF grant DMS-1400073. a r X i v : . [ m a t h . C O ] M a y SERGEY NORIN, YUE RU SUN, AND YI ZHAO
Theorem 1.3.
For all n ≥ m ≥ , (1) r ( S ( n, m )) ≥ m + 53 n + o ( m ) . Further, for n ≥ m , (2) r ( S ( n, m )) ≥ m + 189115 n + o ( m ) . Note that the bounds in Theorem 1.3 imply that Conjecture 1.2 fails for74 m + o ( m ) ≤ n ≤ m − o ( m ) . Theorem 1.3 also provides a negative answer to a related more general question about Ramseynumbers of trees, which we now discuss. Let T be a tree, and let t and t , with t ≤ t , bethe sizes of the color classes in the 2-coloring of T . Then r ( T ) ≥ t + t −
1. Indeed, onecan color the edges of K t + t − in two colors so that the edges of the first color induce thecomplete bipartite graph K t + t − ,t − . Similarly, we have r ( T ) ≥ t − K t − with the first color inducing the complete bipartite graph K t − ,t − . Let r B ( T ) := max(2 t + t − , t − r ( T ) = r B ( T ) forevery tree T . Grossman, Harary and Klawe [8] disproved Burr’s conjecture, by showing thatthe Ramsey number of some double stars is larger than r B ( T ) by one. (See Theorem 1.1.)They asked whether the difference r ( T ) − r B ( T ) can be arbitrarily large. Haxell, (cid:32)Luczak andTingley proved that Burr’s conjecture is asymptotically true for trees with relatively smallmaximum degree. Theorem 1.4 (Haxell, (cid:32)Luczak and Tingley [10]) . For every η > there exists δ > satisfying the following. If T is a tree with maximum degree at most δ | V ( T ) | then r ( T ) ≤ (1 + η ) r B ( T ) . Finally, Erd˝os, Faudree, Rousseau and Schelp [6] asked whether r ( T ) = r B ( T ) for trees T with colors classes of sizes | V ( T ) | / | V ( T ) | /
3. (Note that in the case the two quantitiesin the definition of r B ( T ) are equal, and that Theorem 1.1 does not cover this case for doublestar.) Theorem 1.3 gives a negative answer to this question and to the above question ofGrossman, Harary and Klawe by showing that r ( T ) and r B ( T ) can differ substantially evenfor trees with colors classes of sizes k and 2 k . Indeed, if T = S (2 k − , k −
1) we have r B ( T ) = 4 k −
1, but r ( T ) ≥ . k − o ( k ) by (2).Let us now return to upper bounds. Using Razborov’s flag algebra method, we extend theresults of Theorem 1.1 showing the following. Theorem 1.5. r ( S ( n, m )) ≤ n + 2 m + 2 for m ≤ n ≤ . m + 1) . The paper is structured as follows. In Section 2 we show that the problem of finding r ( S ( n, m )) is essentially equivalent to the problem of characterizing the set of pairs ( δ, η )such that there exists graph G with minimum degree at least δ | V ( G ) | , in which every twovertices have at least η | V ( G ) | common non-neighbors. (See Theorem 2.4 for the precisestatement.) In Section 3 we analyze this set of pairs. In Section 4 we continue the discussionof Ramsey numbers of double stars and prove the consequences of the results of Section 2 inthis context. In particular, we prove Theorems 1.3 and 1.5 and establish general asymptotic SYMPTOTICS OF RAMSEY NUMBERS OF DOUBLE STARS 3 upper and lower bounds on Ramsey numbers of double stars, which differ by less than 2%.(See Theorem 4.5).The paper uses standard graph theoretic notation. In particular, N ( v ) denotes the neigh-borhood of a vertex v in a graph G , when the graph is understood from context.2. From Ramsey numbers to degree conditions
In this section we prove preliminary results which allow us to break the symmetry betweencolors and replace the original Ramsey-theoretic problem by an equivalent problem withTur´an-type flavor. Let (
B, R ) be a partition of the edges of K p into two color classes B and R . For brevity we will say that ( B, R ) is ( n, m ) -free if K p contains no S ( n, m ) with all theedges belonging to the same part of ( B, R ). For v ∈ [ p ] and C ∈ { B, R } , let N C ( v ) denotethe set of vertices joined to v by edges in C , and let deg C ( v ) = | N C ( v ) | .The first lemma that we need is due to Grossman, Harary and Klawe, but we include aproof for completeness. Lemma 2.1 ([8, Lemma 3.4]) . Let p ≥ n + 2 m + 2 , and let ( B, R ) be an ( n, m ) -free partitionof the edges of K p . Then deg C ( v ) ≤ n + m for every v ∈ [ p ] and C ∈ { B, R } .Proof. Choose v ∈ [ p ] and C ∈ { B, R } such that deg C ( v ) is maximum. Suppose for acontradiction that deg C ( v ) ≥ n + m + 1. We assume without loss of generality that C = B .If deg B ( u ) ≥ m + 1 for some u ∈ N B ( v ) then K p contains a double star S ( n, m ) with edgesin B and bridge uv . Thus deg R ( u ) ≥ p − m − ≥ m + n + 1 for every u ∈ N B ( v ). It followsthat there exist u, w ∈ N B ( v ) such that uw ∈ R . In this case ( B, R ) contains a double star S ( n, m ) with edges in R and the bridge uw , a contradiction. (cid:3) Lemma 2.2.
Let p ≥ n + 2 m + 2 , and let ( B, R ) be a partition of the edges of K p . Then ( B, R ) is ( n, m ) -free if and only if for every C ∈ { B, R } and every uv ∈ C either (3) | N C ( u ) ∪ N C ( v ) | ≤ n + m + 1 or (4) deg C ( u ) ≤ n and deg C ( v ) ≤ n. Proof.
Clearly, if uv ∈ C satisfies either (3) or (4) then uv is not a bridge of a monochromatic S ( n, m ). Conversely, suppose that uv ∈ C for some C ∈ { B, R } violates both (3) and (4).In particular, we may assume that deg C ( u ) ≥ n + 1. By Lemma 2.1, we may further assumethat deg C ( v ) ≥ p − n − m − ≥ m + 1. Thus ( B, R ) contains a double star S ( n, m )with edges in C and a bridge uv . (It can be constructed by first choosing n neighbors of u from N C ( u ) ∪ N C ( v ) which will serve as the leaves of S ( n, m ) adjacent to u . We choosethese neighbors outside of N C ( v ) whenever possible. Then at least m elements of N C ( v ) willremain, and can serve as the leaves of S ( n, m ) adjacent to u .) (cid:3) The next key lemma will allow us to break the symmetry between colors and replace theoriginal Ramsey-theoretic problem by an equivalent problem with Tur´an-type flavor.
Lemma 2.3.
Let p ≥ max(2 n + 2 , n + 2 m + 2) , and let ( B, R ) be an ( n, m ) -free partition ofthe edges of K p . Then there exists C ∈ { B, R } such that deg C ( v ) ≤ n for all v ∈ [ p ] .Proof. Suppose for a contradiction that there exists v ∈ [ p ] such that deg B ( v ) ≥ n + 1, and v ∈ [ p ] such that deg R ( v ) ≥ n + 1. Then, as p ≥ n + 2, there exists a partition ( V B , V R ) SERGEY NORIN, YUE RU SUN, AND YI ZHAO of [ p ] such that deg B ( v ) ≥ n + 1 for every v ∈ V B , deg R ( v ) ≥ n + 1 for every v ∈ V R , and V B , V R (cid:54) = ∅ . As (
B, R ) is ( n, m )-free it follows from Lemma 2.2 that(5) | N C ( u ) ∩ N C ( v ) | ≥ p − n − m − u ∈ V B , v ∈ V R , C ∈ { B, R } such that uv (cid:54)∈ C Let b : [ p ] → { , } be the characteristic function of B , that is b ( uv ) = 1 if and only if { u, v } ∈ B . Define r : [ p ] → { , } analogously. Then (5) can be rewritten as(6) (cid:88) w ∈ [ p ] ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw )) ≥ p − n − m − u ∈ V B , v ∈ V R . Summing (6) over all such pairs u and v we obtain (cid:88) ( u,w ) ∈ V B ,v ∈ V R ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw ))+ (cid:88) u ∈ V B , ( v,w ) ∈ V R ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw )) ≥ ( p − n − m − | V B || V R | (7)On the other hand for every v ∈ V R ,(8) (cid:88) ( u,w ) ∈ V B ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw )) = | N B ( v ) ∩ V B || N R ( v ) ∩ V B | ≤ | V B | Similarly for every u ∈ V B ,(9) (cid:88) ( v,w ) ∈ V R ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw )) ≤ | V R | Thus (cid:88) ( u,w ) ∈ V B ,v ∈ V R ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw ))+ (cid:88) u ∈ V B , ( v,w ) ∈ V R ( b ( uv ) r ( uw ) r ( vw ) + r ( uv ) b ( uw ) b ( vw )) ≤
14 ( | V R || V B | + | V B || V R | )(10)Combining (7) and (10) we obtain(11) p − n − m − ≤
14 ( | V R | + | V B | ) = p . Inequality (11) can be rewritten as 3 p ≤ m + 4 n + 4. However,3 p ≥ n + 2 m + 2) + (2 n + 2) = 4 m + 4 n + 6 , implying the desired contradiction. (cid:3) Lemma 2.3 readily implies the following main result of this section.
Theorem 2.4.
Let n ≥ m ≥ and p ≥ max(2 n + 2 , n + 2 m + 2) be integers. Then thefollowing are equivalent (i) p < r ( S ( n, m )) , SYMPTOTICS OF RAMSEY NUMBERS OF DOUBLE STARS 5 (ii) there exists a graph G with | V ( G ) | = p such that deg( v ) ≥ p − n − for every v ∈ V ( G ) and | N ( v ) ∪ N ( u ) | ≤ n + m + 1 for all uv ∈ E ( G ) .Proof. (i) ⇒ (ii). Let (
B, R ) be an ( n, m )-free partition of the edges of K p . By Lemma 2.3,we assume without loss of generality that deg R ( v ) ≤ n for every v ∈ [ p ], or equivalentlydeg B ( v ) ≥ p − n − ≥ n + 1. Let G be the graph with V ( G ) = [ p ] and E ( G ) = B . ByLemma 2.2 | N ( v ) ∪ N ( u ) | ≤ n + m + 1 for all uv ∈ E ( G ). Thus G satisfies (ii). (ii) ⇒ (i). Let (
B, R ) be a partition of the edges of the complete graph with the vertexset V ( G ) such that B = E ( G ). Then neither B nor R contains the edge set of a double star S ( n, m ) by Lemma 2.2. Thus p < r ( S ( n, m )). (cid:3) Valid points
By Theorem 2.4, the function r ( S ( n, m )) is completely determined by the answer to thefollowing question: For which triples ( p, d, s ) does there exist a graph G with | V ( G ) | = p such that deg( v ) ≥ d for every v ∈ V ( G ) and | N ( v ) ∪ N ( u ) | ≤ s for all uv ∈ E ( G )?We will be primarily interested in the asymptotic behavior of r ( S ( n, m )), and thus ratherthan answering the (likely very difficult) question above we analyze the following setting.Given 0 ≤ δ, η ≤ G with | V ( G ) | > δ, η ) -graph if • deg( v ) + 1 ≥ δ | V ( G ) | for every v ∈ V ( G ), and • | N ( v ) ∪ N ( u ) | ≤ (1 − η ) | V ( G ) | for all uv ∈ E ( G ).We say that ( δ, η ) ∈ [0 , is directly valid if there exists a ( δ, η )-graph, and we say that( δ, η ) ∈ [0 , is valid if it belongs to the closure of the set of directly valid points. Let V ⊆ [0 , denote the set of valid points. Note, in particular, that if 0 ≤ x ≤ x , 0 ≤ y ≤ y and ( x , y ) ∈ V then ( x , y ) ∈ V . Finally, a point ( δ, η ) ∈ [0 , is invalid if it is not valid.In this section we approximate the set of valid points. Lemma 3.1.
For n, δ, η ≥ , let G be a ( δn ) -regular graph with | V ( G ) | = n such that | N ( v ) ∪ N ( u ) | ≤ (1 − η ) n for all uv ∈ E ( G ) . Then (cid:18) n + pδ, − n − (cid:18) δ − n (cid:19) p + (2 δ + η − p (cid:19) ∈ V for every p ∈ [0 , .Proof. We will construct a “random sparsified blow-up” of G as follows. Let k be an integer,let U be a set with | U | = kn , and let φ : U → V ( G ) be a map such that | φ − ( v ) | = k forevery v ∈ V ( G ). Let G (cid:48) be a random graph with V ( G (cid:48) ) = U is constructed as follows. Let uv ∈ E ( G (cid:48) ) if φ ( u ) = φ ( v ), let uv (cid:54)∈ E ( G (cid:48) ) if φ ( u ) (cid:54) = φ ( v ) and φ ( u ) φ ( v ) (cid:54)∈ E ( G ), and finallylet uv be an edge of G with probability p (independently for each edge) if φ ( u ) φ ( v ) ∈ E ( G ).(It is natural to think of G (cid:48) as a graph obtained from G by replacing every vertex by a cliqueof size k and every edge by a random bipartite graph with density p .)We have almost surely deg( v ) ≥ (1 + pδn ) k − o ( k ) for each v ∈ V ( G (cid:48) ). Furthermore, let uv ∈ E ( G (cid:48) ) be such that φ ( v ) (cid:54) = φ ( u ), and let η (cid:48) = | N ( φ ( v )) ∩ N ( φ ( u )) | /n , then 2 δ − η (cid:48) ≤ − η ,and almost surely nk − | N ( v ) ∪ N ( u ) | ≥ kn (1 − δ + η (cid:48) + 2( δ − η (cid:48) − /n )(1 − p ) + η (cid:48) (1 − p ) ) + o ( k )= kn (1 − /n − δ − /n ) p + η (cid:48) p ) ≥ kn (1 − /n − δ − /n ) p + (2 δ + η − p )Thus G (cid:48) is almost surely a (1 /n + pδ − o (1) , − /n − δ − /n ) p +(2 δ + η − p − o (1))-graph.It follows that (1 /n + pδ, − /n − δ − /n ) p + (2 δ + η − p ) ∈ V . (cid:3) SERGEY NORIN, YUE RU SUN, AND YI ZHAO
Corollary 3.2.
For every p ∈ [0 , , (cid:18) p , − p (cid:19) , (cid:18) p , − p + 5 p (cid:19) ∈ V Proof.
We apply Lemma 3.1 to the cycle of length five ( n = 5, δ = 2 / η = 1 / n = 21, δ = 10 / η = 6 / (cid:3) We use Corollary 3.2 to approximate V from below. Approximating V from above requiresthe use of flag algebras. i δ ∗ i η ∗ i Table 1.
Invalid pairs ( δ ∗ i , η ∗ i ). Theorem 3.3.
The pairs ( δ ∗ i , η ∗ i ) for ≤ i ≤ given in Table 1 are invalid.Proof. The proof is computer-generated and consists of a flag algebra computation carriedout in Flagmatic [16]. It is accomplished by executing the following script, which pro-duces certificates of infeasibility that can be found at . from flagmatic.all import *p = GraphProblem(7, density=[("3:121323",1),("3:",1)], mode="optimization")p.add_assumption("1:",[("2:12(1)",1)],$\delta^*_i$)p.add_assumption("2:12",[("3:12(2)",1)],$\eta^*_i$)p.solve_sdp(show_output=True, solver="csdp") As the usage of flag algebra computations to obtain similar bounds has become standard inthe area in recent years (see a survey [15]), and the method is described in great detail ina number of papers (e.g. [5, 11, 12, 13]), we avoid extensive discussion of the flag algebrasetting. Essentially, nonexistence of ( δ ∗ i − ε, η ∗ i − ε )-graphs for some positive ε is provedby exhibiting a system of inequalities, involving homomorphism densities of seven vertexgraphs, which has to hold in every ( δ ∗ i − ε, η ∗ i − ε )-graph, but which has no solutions. (cid:3) As we will see in Section 4 for the purposes of investigating Ramsey numbers we areprimarily interested in the restriction of V to the region [0 . , . × [0 . , . SYMPTOTICS OF RAMSEY NUMBERS OF DOUBLE STARS 7
Figure 1.
The restrictions of the valid (blue) and invalid (red) point sets tothe rectangle [0 . , . × [0 . , . Theorem 3.4.
For every ε > the pair (1 / ε, / ε ) is invalid.Proof. It suffices to show that, if G is a graph with | V ( G ) | = n such that deg( v ) > n/ v ∈ V ( G ), then there exists an edge uv ∈ V ( G ) such that | N ( v ) ∪ N ( u ) | > n/ u, w ∈ V ( G ), there exists v ∈ V ( G )such that uv, wv ∈ E ( G ). It follows that | N ( u ) ∩ N ( w ) | ≥ | N ( u ) ∩ N ( w ) ∩ N ( v ) | ≥ | N ( u ) ∩ N ( v ) | + | N ( w ) ∩ N ( v ) | − | N ( v ) | = ( | N ( u ) | + | N ( v ) | − | N ( u ) ∪ N ( v ) | ) + ( | N ( w ) | + | N ( v ) | − | N ( w ) ∪ N ( v ) | ) − | N ( v ) |≥ | N ( v ) | + | N ( u ) | + | N ( w ) | − · n > n . (12) SERGEY NORIN, YUE RU SUN, AND YI ZHAO
Therefore, n ≥ (cid:88) u ∈ V ( G ) deg( u )( n − deg( u ))= (cid:88) u ∈ V ( G ) ,v ∈ N ( u ) ,w (cid:54)∈ N ( u ) (cid:88) ( u,w ) ∈ V ( G ) uw (cid:54)∈ E ( G ) | N ( u ) ∩ N ( w ) | + (cid:88) ( u,v ) ∈ V ( G ) uv ∈ E ( G ) n − | N ( u ) ∩ N ( v ) | (12) > n n − | E ( G ) | ) + n · | E ( G ) | = n n | E ( G ) | ≥ n , a contradiction, as desired. (cid:3) Back to Ramsey numbers
In this section we derive bounds on Ramsey numbers of double stars from the informationon the set of valid points obtained in Section 3. In particular we prove Theorems 1.3 and 1.5.Our first lemma follows immediately from the definition of a directly valid point andTheorem 2.4.
Lemma 4.1.
Let n ≥ m ≥ be integers, and let p = r ( S ( n, m )) − . If p ≥ max(2 n + 2 , n +2 m + 2) then (cid:18) − np , − n + m + 1 p (cid:19) is directly valid. The next corollary is in turn a direct consequence of Lemma 4.1.
Corollary 4.2.
Let n ≥ m ≥ be integers, and let ( δ, η ) be an invalid point then r ( S ( n, m )) ≤ max (cid:18) n + 2 , n + 2 m + 2 , (cid:24) n − δ (cid:25) , (cid:24) n + m + 11 − η (cid:25)(cid:19) . We are now ready to derive Theorem 1.5.
Proof of Theorem 1.5.
By Theorem 3.3 the point (0 . , . n ≤ . m +1) we have n − . ≤ n + 2 m + 2 and n + m + 11 − . ≤ n + 2 m + 2 . Thus r ( S ( n, m )) ≤ n + 2 m + 2 by Corollary 4.2. (cid:3) Next we turn to asymptotic bound on r ( S ( n, m )). For x ∈ [1 , + ∞ ) defineˆ r ( x ) := lim n,m →∞ n/m → x r ( S ( n, m )) m . The next theorem expresses ˆ r ( x ) in terms of V . SYMPTOTICS OF RAMSEY NUMBERS OF DOUBLE STARS 9
Theorem 4.3.
For every x ≥ , let ˆ r (cid:48) ( x ) = max (cid:26) r : (cid:18) − xr , − x + 1 r (cid:19) ∈ V (cid:27) . Then ˆ r ( x ) = max(2 x, x + 2 , ˆ r (cid:48) ( x )) . In particular, the limit in the definition of ˆ r ( x ) exists.Proof. It follows immediately from Corollary 4.2 thatlim sup n,m →∞ n/m → x r ( S ( n, m )) m ≤ max(2 x, x + 2 , ˆ r (cid:48) ( x )) . Since r ( S ( n, m )) ≥ max(2 n + 2 , n + 2 m + 2) for n ≥ m ≥
3, it remains to show that for all x ≥ ε > γ, N > m ≥ N n ≥ ( x − γ ) m are integers, then r ( S ( n, m )) ≥ (ˆ r (cid:48) ( x ) − ε ) m . Let r = ˆ r (cid:48) ( x ) , δ = 1 − xr − ε r , and η = 1 − x + 1 r − ε r . The point ( δ, η ) is directly valid by definition of V . Therefore there exists a graph G with | V ( G ) | > v ) + 1 ≥ δ | V ( G ) | for every v ∈ V ( G ) and | N ( v ) ∪ N ( u ) | ≤ (1 − η ) | V ( G ) | for all uv ∈ E ( G ). Let s = | V ( G ) | , γ = ε r , N = (cid:100) s/ε (cid:101) . Let m ≥ N , n ≥ ( x − γ ) m be integers, and let k = (cid:98) ( r − ε/ m/s (cid:99) . Then(13) (1 − δ ) ks ≤ (cid:16) xr + ε r (cid:17) (cid:16) r − ε (cid:17) m ≤ (cid:18) x − ε r (cid:19) m ≤ n. Similarly,(14) (1 − η ) ks ≤ (cid:18) x + 1 r + ε r (cid:19) (cid:16) r − ε (cid:17) m ≤ (cid:18) x + 1 − ε r (cid:19) m ≤ m + n. Let G (cid:48) be the graph with | V ( G ) | = ks obtained by replacing every vertex of G by a completegraph on k vertices and replacing all the edges by complete bipartite graphs. By (13) and(14), the graph G (cid:48) satisfies the conditions in Theorem 2.4(ii). Therefore r ( S ( n, m )) ≥ ks = (cid:98) ( r − ε/ m/s (cid:99) s ≥ ( r − ε/ m − s ≥ ( r − ε ) m by the choice of N , as desired. (cid:3) Corollary 4.4.
The following inequalities hold: ˆ r ( x ) ≥ x + 56 for 1 ≤ x, (15) ˆ r ( x ) ≥ x for 1 ≤ x ≤ , (16) ˆ r ( x ) ≥ x + 2123 for 2 ≤ x. (17) Proof.
For x ≥
1, let p = ( x + 2) / (2 x + 1) ≤
1, and let r = 5 x/ /
6. Direct computationsshow that 1 − x/r = (1 + 2 p ) /
5, and 1 − ( x + 1) /r = (3 − p ) /
5. By Corollary 3.2,((1 + 2 p ) / , (3 − p ) / ∈ V . Thus ˆ r ( x ) ≥ r by Theorem 4.3, and (15) holds. For the proof of (16), let r = 21 x/
10. Then 1 − x/r = 11 /
21, and 1 − ( x + 1) /r ≤ / ≤ x ≤
2. We have (11 / , / ∈ V by Corollary 3.2 applied with p = 1. Thus (16)holds.Finally, for the proof of (17), let x ≥
2, let p = (20 + 13 x ) / (10 + 18 x ) ≤
1, and let r = 189 x/
115 + 21 /
23. Then 1 − x/r = (1 + 10 p ) /
21 and1 − x + 1 r = 19 − p + 5 p − (cid:18) x −
25 + 9 x (cid:19) ≤ − p + 5 p . By Corollary 3.2, we have ((1 + 10 p ) / , (19 − p + 5 p ) / ∈ V , implying ˆ r ( x ) ≥ r byTheorem 4.3. (cid:3) Let us note that the lower bound in (17) can be tightened toˆ r ( x ) ≥ (cid:16) x + √
25 + 40 x + 106 x (cid:17) . However, we chose to keep the bound linear, and hence hopefully more transparent.Theorem 1.3 follows immediately from Corollary 4.4.
Proof of Theorem 1.3.
Note that r ( S ( n, m )) = ˆ r ( nm ) m + o ( m ) . Thus inequalities (1) and (2) Theorem 1.3 follow from inequalities (15) and (17) in Corol-lary 4.4, respectively. (cid:3)
More generally, Corollary 4.4 implies the following piecewise linear lower bound on ˆ r ( x ).Let ˆ r l ( x ) = x + 2 for 1 ≤ x ≤ x + for ≤ x ≤ , x for ≤ x ≤ , x + for 2 ≤ x ≤ , x for ≤ x. Coming back to upper bounds, define u δ,η ( x ) = max (cid:18) x + 2 , x, x − δ , x + 11 − η (cid:19) for ( δ, η ) ∈ [0 , . It follows from Theorem 4.3 that ˆ r ( x ) ≤ u δ,η ( x ) for every point ( δ, η ) thatlies in the closure of the set of invalid points. Accordingly we defineˆ r u ( x ) = min i =1 u δ ∗ i ,η ∗ i ( x ) , where the pairs ( δ ∗ i , η ∗ i ) for 1 ≤ i ≤ δ ∗ , η ∗ ) = (1 / , / r u ( x ) is an upper bound on ˆ r ( x ). Next theoremcollects all the asymptotic lower and upper bounds on Ramsey numbers of double starsestablished in this section. Theorem 4.5. ˆ r l ( x ) ≤ ˆ r ( x ) ≤ ˆ r u ( x ) . SYMPTOTICS OF RAMSEY NUMBERS OF DOUBLE STARS 11
Figure 2.
Functions ˆ r ∗ l ( x ),ˆ r l ( x ) and ˆ r u ( x ) on an interval 1 . ≤ x ≤ . Figure 3.
The ratio ˆ r u ( x ) / ˆ r l ( x ) on an interval 1 . ≤ x ≤ r ∗ l ( x ) = max( x + 2 , x ) equal to thevalue of ˆ r ( x ) conjectured in [8]. Functions ˆ r ∗ l ( x ),ˆ r l ( x ) and ˆ r u ( x ) are plotted in Figure 2.The ratio ˆ r u ( x ) / ˆ r l ( x ) is plotted in Figure 3. In particular, the bounds in Theorem 4.5asymptotically predict the value of r ( S ( n, m )) with the error less that 1%. In comparison, asmentioned in the introduction, the value of r ( S (2 m, m )) conjectured in [8] is asymptoticallysmaller than the lower bound provided in Theorem 1.3 by 5%.5. Concluding remarks
Asymptotic value of r ( S ( n, m )) . The constructions we used to provide the new lowerbounds on Ramsey numbers of double stars are not simple, and we do not attempt toconjecture their tightness. Understanding the asymptotic behavior of r ( S (2 m, m )) appearsto be difficult already. Question 5.1. Is r ( S (2 m, m )) = 4 . m + o ( m ) ? Perhaps, a combination of flag algebra techniques with stability methods, along the linesof the arguments in [1, 2], can be used to resolve the above question. More ambitiously, onecan ask the following.
Question 5.2.
Let T be a tree on n vertices. Suppose that color classes in the 2-coloring of T have sizes n/ and n/ . Is r ( T ) ≤ . n + o ( n ) ? When is r ( S ( n, m )) = 2 n + 2 ? In Theorem 1.5 we were able to substantially extend therange of known values ( m, n ) for which the equality r ( S ( n, m )) = n + 2 m + 2 holds. We werenot similarly successful in reducing the lower bound on n in Theorem 1.1 which guarantees r ( S ( n, m )) = 2 n + 2. By Theorem 2.4, finding the optimal bound is essentially equivalentto answering the following question. Question 5.3.
Find the infimum c inf of the set of real numbers c for which there exists agraph G with | V ( G ) | = n such that • deg( v ) > n/ for every v ∈ V ( G ) , and • | N ( v ) ∪ N ( u ) | ≤ cn for every uv ∈ E ( G ) . Theorem 3.4 shows that c inf ≥ /
3. The sparsified blow-ups of the line graph of K ,introduced in the proof of Lemma 3.1, show that c inf ≤ / ≈ . c inf > /
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