aa r X i v : . [ m a t h . C O ] F e b Ramsey NumbersforNon-trivial Berge Cycles
Jiaxi Nie ∗ Jacques Verstra¨ete † Department of MathematicsUniversity of California, San Diego9500 Gilman DriveLa Jolla CA 92093-0112.February 9, 2021
Abstract
In this paper, we consider an extension of cycle-complete graph Ramsey numbersto Berge cycles in hypergraphs: for k ≥
2, a non-trivial Berge k -cycle is a family of sets e , e , . . . , e k such that e ∩ e , e ∩ e , . . . , e k ∩ e has a system of distinct representativesand e ∩ e ∩ · · · ∩ e k = ∅ . In the case that all the sets e i have size three, let B k denotesthe family of all non-trivial Berge k -cycles. The Ramsey numbers R ( t, B k ) denote theminimum n such that every n -vertex 3-uniform hypergraph contains either a non-trivialBerge k -cycle or an independent set of size t . We prove R ( t, B k ) ≤ t k − + √ log t and moreover, we show that if a conjecture of Erd˝os and Simonovits [11] on girth ingraphs is true, then this is tight up to a factor t o (1) as t → ∞ . ∗ E-mail: [email protected] † Research supported by NSF Award DMS-1952786. E-mail: [email protected] Introduction
Let F be a family of r -graphs and t ≥
1. The Ramsey numbers R ( t, F ) denote the minimum n such that every n -vertex r -graph contains either a hypergraph in F or an independent setof size t . For k ≥
2, a
Berge k -cycle is a family of sets e , e , . . . , e k such that e ∩ e , e ∩ e , . . . , e k ∩ e has a system of distinct representatives, and a Berge cycle is non-trivial if e ∩ e ∩ · · · ∩ e k = ∅ . Let B rk denote the family of Berge k -cycles all of whose sets have size r . When r = 2, B k = { C k } , where C k denotes the graph cycle of length k . In this paper, welet B k = B k .It is a notoriously difficult problem to determine even the order of magnitude of R ( t, C k ) –the cycle-complete graph Ramsey numbers. Kim [17] proved R ( t, C ) = Ω( t / log t ), whichgives the order of magnitude of R ( t, C ) when combined with the results of Ajtai, Koml´osand Szemer´edi [1] and Shearer [26]. The current state-of-the-art results on R ( t, C ) are dueto Fiz Pontiveros, Griffiths and Morris [12] and Bohman and Keevash [5], using the randomtriangle-free process, which determines R ( t, C ) up to a small constant factor. The case R ( t, C ) is the subject of a notorious conjecture of Erd˝os [6]. The current best upper boundson R ( t, C k ) come from the work of Caro, Li, Rousseau and Zhang [8] when k is even andSudakov [27] when k is odd. Recent results using pseudorandom graphs by Mubayi and thesecond author [23] give the best lower bounds on cycle-complete graph Ramsey numbers.For k ≥
3, a loose k -cycle is a non-trivial Berge k -cycle, denoted C rk , with sets e , e , . . . , e k of size r such that | e ∩ e | = 1, | e ∩ e | = 1 , . . . , | e k ∩ e | = 1, and for any other pairsof edges e i , e j , e i ∩ e j = ∅ . Ramsey type problems for loose cycles in r -graphs have beenstudied extensively [3, 9, 10, 13, 15–19, 21, 23]. For r -uniform hypergraphs, or simple r -graphs ,with r ≥
3, Kostochka, Mubayi and the second author [18] proved for all r ≥
3, there existconstants a, b > at (log n ) ≤ R ( t, C r ) ≤ bt , (1)The following conjecture was proposed in [18]: Conjecture I.
For r, k ≥ , R ( t, C rk ) = t kk − + o (1) . (2)The conjecture is true for k = 3 due to (1). It is shown in [25] that R ( t, C ) ≤ t / o (1) ,and M´eroueh [21] showed R ( t, C rk ) = O ( t / ⌊ ( k +1) / ⌋ ) for k ≥
3, improving earlier results ofCollier-Cartaino, Graber and Jiang [9]. Conjecture I motivates our current study of non-trivial Berge k -cycles. In support the above conjecture, we prove the following result for2on-trivial Berge cycles of even length: Theorem 1.
For k ≥ , R ( t, B k ) ≤ t k k − + √ log t . Erd˝os and Simonovits [11] conjectured that there exists an n -vertex graph of girth more than2 k with Θ( n /k ) edges. This notoriously difficult conjecture remains open, except when k ∈ { , , } , largely due to the existence of generalized polygons [2, 28, 29]. We prove thefollowing theorem relating this conjecture to lower bounds on Ramsey numbers for non-trivialBerge cycles: Theorem 2.
Let k ≥ . Suppose there exists an n -vertex graph of girth more than k with c n /k edges for some positive constant c . Then for t large enough and some positiveconstant c , R ( t, B k ) ≥ t kk − − c t log t . (3)This shows that if the Erd˝os-Simonovits Conjecture is true, then Theorem 1 is tight up to a t o (1) factor.We prove Theorem 1 in Section 4 and Theorem 2 in Section 2. Theorem 2 is valid for oddvalues of k , and we believe Theorem 1 should extend to odd values of k and all r ≥ Conjecture II.
For all r, k ≥ , R ( t, B rk ) ≤ t kk − + o (1) . (4) Notation and terminology.
For a hypergraph H , let V ( H ) denote the vertex set of H , v ( H ) = | V ( H ) | and let | H | be the number of edges in H . If all edges of H havesize r , we say H is an r -uniform hypergraph , or an r -graph for short. For v ∈ V ( H ), let d H ( v ) = |{ e ∈ H : v ∈ e }| be the degree of v in H . We denote the average degree of H by d av ( H ), denote the minimum degree of H by d min ( H ), and the maximum degree of H by d max ( H ). For u, v ∈ V ( H ), let d H ( u, v ) = |{ w : uvw ∈ H }| denote the codegree of thepair { u, v } . An independent set in a hypergraph is a set of vertices containing no edge of thehypergraph. Let α ( H ) denote the largest size of an independent set in a hypergraph H . We will use the following lemma to get a large bipartite subgraph with large minimum degreeand small maximum degree: 3 emma 3.
Let k ≥ , c > , and let G be an n -vertex graph of girth more than k with morethan cn /k edges. Then there exists a bipartite subgraph G ′ of G such that d min ( G ′ ) ≥ cn /k , d max ( G ′ ) ≤ n /k /c k − , and v ( G ′ ) ≥ c k n .Proof. A maximum cut of G gives a bipartite subgraph with at least cn /k edges. Asubgraph G ′ of this bipartite subgraph of minimum degree at least cn /k + 1 may be ob-tained by repeatedly removing vertices of degree at most cn /k . If G ′ has maximum degree d max ( G ′ ) = d max , and v is a vertex of maximum degree in G ′ , then the number of vertices atdistance k from v is at least d max c k − n ( k − /k , since G has girth larger than 2 k . In particular, d max c k − n ( k − /k ≤ n and so d max ≤ n /k /c k − . The number of vertices in G ′ is at least c k n ,since G ′ has minimum degree at least cn /k + 1 and girth larger than 2 k .Let r ≥
2, a star with vertex set V is an r -graph on V consisting of all edges containingsome vertex of V , i.e., the edge set of a star is { e ⊂ V : | e | = r, v ∈ e } for some vertex v ∈ V . Lemma 4.
Let integer r ≥ , and let integers d ≥ m . Let S d,m be a d -vertex r -graphconsisting of m vertex-disjoint stars of size ⌊ d/m ⌋ or ⌈ d/m ⌉ . Then the probability that auniformly chosen set of s vertices of S d,m is independent is at most exp( − m ( s − rm )2 d ) . Proof.
Let the vertex set of these stars be V , V , . . . , V m . The probability that a uniformlychosen set of s i vertices in V i is independent in S d,m is at most 1 − s i / ⌈ d/m ⌉ ≤ − ms i / d if s i ≥ r , and is 1 if s i < r . Hence, this probability is at most 1 − m ( s i − r ) / d for 0 ≤ s i ≤ d .Therefore a uniformly chosen set I ⊂ S d,m of s vertices with | I ∩ V i | = s i is independent withprobability at most m Y i =1 (cid:18) − m ( s i − r )2 d (cid:19) ≤ exp − m X i =1 m ( s i − r )2 d ! = exp (cid:18) − m ( s − rm )2 d (cid:19) Since this upper bound is independent of the s i , this gives the lemma.Now we are ready to prove Theorem 2. Proof of Theorem 2.
It suffices to show that for n large enough, there exists an n -vertex B k -free triple system with independence number O ( n − k log n ). Let G be an n -vertex graphof girth more than 2 k with 2 cn /k edges for some positive constant c . By Lemma 3, thereexists a bipartite subgraph G ′ of G with at least N = c k n vertices, minimum degree at least4 n /k and maximum degree at most n /k /c k − . Let X, Y be the parts of this bipartite graphwhere | Y | ≥ | X | . Let m = 8 log n/c k . We form an r -graph H with vertex set Y by placing arandom copy of S d ( x ) ,m on the vertex set N G ( x ), the neighborhood of x in G , independentlyfor each x ∈ X . Since G has girth more than 2 k , it is straightforward to check that H does not contain any non-trivial Berge k -cycle. We now compute the expected number ofindependent set of size t = 3 mn − /k /c k +1 in H . If it is possible that H has no independentset of size t , then since v ( H ) ≥ N/
2, we find that R ( t, B k ) ≥ N/ ≥ t kk − − ck log log t log t , for some positive constant c k . This is enough to prove Theorem 2.For an independent t -set I in H , I ∩ N G ( x ) is an independent set in S d ( x ) ,m for all x ∈ X .Since these events are independent, setting s ( x ) = | I ∩ N G ( x ) | , Lemma 4 gives: P ( I independent in H ) ≤ Y x ∈ X exp (cid:18) − m ( s ( x ) − rm )2 d ( x ) (cid:19) = exp − X x ∈ X ms ( x )2 d ( x ) + X x ∈ X rm d ( x ) ! . For every x ∈ X , cn /k ≤ d ( x ) ≤ n /k /c k − and therefore P ( I independent in H ) ≤ exp (cid:18) − c k − m P x ∈ X s ( x )2 n /k + | X | rm cn /k (cid:19) . Now P x ∈ X s ( x ) is precisely the number of edges of G ′ between X and I . Since every vertexin I has degree at least cn /k , this number is at least cn /k t = rmn/c k . Consequently, using | X | < n/ P ( I independent in H ) ≤ exp (cid:18) − c k mt c k mt (cid:19) = exp (cid:18) − c k mt (cid:19) . The expected number of independent set of size t is at most (cid:18) nt (cid:19) exp (cid:18) − c k mt (cid:19) < exp (cid:18) t log n − c k mt (cid:19) = exp ( − t log n ) . This is vanishing as n → ∞ , and the proof of Theorem 2 is complete.5 Degrees, codegrees and independent sets
Let α ( H ) be the size of a largest independent set in a hypergraph H . We make use of thefollowing elementary lemma, whose proof is a standard probabilistic argument, included forcompleteness: Lemma 5.
Let d ≥ , and let H be a 3-graph of average degree at most d . Then α ( H ) ≥ v ( H )3 d . Proof.
Let X be a subset of V ( H ) whose elements are chosen independently with probability p = d − / . We can get an independent set by deleting a vertex for each edge of H containedin X . Then the expected size of such independent set is at least pv ( H ) − p | H | = pv ( H ) − p dv ( H )3 = 2 v ( H )3 d . Hence, there must exist an independent set of size larger than the required lower bound,which completes the proof.
Lemma 6.
Let H be a -graph on n vertices, and < ǫ < / . Then there exists an inducedsubgraph G satisfying the following properties:1. v ( G ) ≥ n − ǫ ) ,2. d max ( G ) ≤ d av ( G ) ǫ .Proof. If d max ( H ) < d av ( H ) /ǫ , then H itself is the desired subgraph. Otherwise, iterativelydelete vertices of H with degree at least d av ( H ). Each deleted vertex will result in the lossof at least d av ( H ) edges. So we can at most delete | H | d av ( H ) = n · d av ( H )3 · d av ( H ) = n < n G (1) be the subgraph induced by the vertices remained. Then we have v ( G (1) ) >n/
2. If d max ( G (1) ) < d av ( G (1) ) /ǫ , then G (1) is the desired subgraph. Otherwise, we have d av ( G (1) ) ≤ ǫd max ( G (1) ) < ǫd av ( H ) . In this case, we repeat the argument above with H replaced by G (1) . Let K = 2 log /ǫ n .We must have obtained an induced subgraph G with d max ( G ) ≤ d av ( G ) /ǫ after at most K ǫ after eachrepetition, the graph remained will have no edge after K repetitions. Suppose after m ≤ K repetitions we have the desired induced subgraph G with d max ( G ) < d av ( G ) /ǫ . Since thenumber of vertices decrease at most by a factor of 2, we also have v ( G ) > n m ≥ n − ǫ ) . This completes the proof.We use the following slightly weaker version of a lemma due to M´eroueh [21]; the lemma isin fact valid for 3-graphs H with no loose k -cycles: Lemma 7.
Let H be a B k -free -graph. Then there exists a subgraph H ∗ of H such that | H ∗ | > | H | /k and each edge of H ∗ contains a pair of codegree 1.Proof. Given a 3-graph G and a pair of vertices x, y , we say that { x, y } is G -heavy if d G ( x, y ) ≥ k , otherwise we say it is G -light . Let G = H , and let H consist of all edges of G containing a G -light pair, and let G = G \ H . For i ≥
2, let H i consist of all edgesof G i containing a G i -light pair, and let G i +1 = G i \ H i . Suppose for contradiction that G k is not empty. Let e = { v , v , v } be an edge in G k , then by definition, { v , v } must be a G k − -heavy pair, and hence, there exists an edge e = { v , v , v } such that v = v . For2 ≤ i ≤ k −
1, let e i = { v i , v i +1 , v i +2 } be an edge in G k +1 − i . By definition, { v i +1 , v i +2 } mustbe a G k − i -heavy pair, and hence, there exists an edge e i +1 = { v i +1 , v i +2 , v i +3 } in G k − i suchthat v i +3 is distinct from all v j , 1 ≤ j ≤ i . Therefore, we have a tight path of length k in G = H , that is, a hypergraph consisted of k + 2 distinct vertices v i , 1 ≤ i ≤ k + 2,and k edges e i = { v i , v i +1 , v i +2 } , 1 ≤ i ≤ k . This is also a non-trivial Berge k-cycle.Noticed that when k is even, { v , v , . . . , v k , v k +1 , v k − , . . . , v } forms a system of distinctrepresentatives of { e ∩ e , e ∩ e , e ∩ e , . . . , e k − ∩ e k , e k ∩ e k − , e k − ∩ e k − , . . . , e ∩ e } , andwhen k is odd, { v , v , . . . , v k +1 , v k , v k − , . . . , v } forms a system of distinct representativesof { e ∩ e , e ∩ e , e ∩ e , . . . , e k − ∩ e k − , e k − ∩ e k , e k ∩ e k − , . . . , e ∩ e } . This results in acontradiction, since H is B k -free. Therefore, G k must be empty, and hence H can be parti-tioned into k − H i , 1 ≤ i ≤ k −
1, such that all H i consist of edges containing an H i -light pair. Let H ′ be the subgraph H i with most edges, then by the pigeonhole principle, | H ′ | > | H | k Now consider a graph J whose vertex set is the set of edges of H ′ , and two edges of H ′ forman edge of J if they share an H ′ -light pair. Not hard to see that J has maximum degree at7ost k −
2. Then by Tur´an’s Theorem, J has an independent set of size at least v ( J ) / ( k − H ′′ of H ′ such that | H ′′ | > | H ′ | k − > | H | k , and each edge of H ′′ contains a pair of codegree 1. A key ingredient of the proof of Theorem 1 is a supersaturation theorem for cycles in graphs:we make use of the following result proved by Simonovits [7] (see Morris and Saxton [22] forstronger supersaturation):
Lemma 8.
For every n, k ≥ , there exist constants δ, b > such that for every b ≥ b ,any n -vertex graph G with at least bn /k edges contains at least δb k n copies of C k . We next give a simple lemma which says that if a graph has many cycles of length 2 k , thenit has many edges. Lemma 9.
Let G be a graph containing m cycles of length k , each containing an edge e ∈ G . Then | G | ≥ m / ( k − / .Proof. For each cycle C of length 2 k containing e , let M ( C ) be the perfect matching of C containing e . Fixing a matching M ⊂ G of size k containing e , at most ( k − k − cycles C have M ( C ) = M . It follows that the number of distinct matching M ⊂ G of size k containing e is at least m/ ( k − k − , and therefore (cid:18) | G | − k − (cid:19) ≥ m ( k − k − . We conclude | G | k − ≥ m/ k − and therefore | G | ≥ m / ( k − / Proof of Theorem 1.
It suffices to show that every n -vertex B k -free triple system H containsan independent set of size at least n (2 k − / (2 k ) − / √ log n . By Lemma 6 with ǫ = exp ( −√ log n ),we find an induced subgraph H of H with n vertices, average degree d and maximumdegree D such that n ≥ n − / √ log n and D < d /ǫ . By Lemma 7, there is a subgraph H of8 with at least | H | / (4 k ) edges such that each edge of H contains a pair of codegree 1 in H . Let χ : V ( H ) → { , , } be a random 3-coloring and let H consist of all triples in H such that the pair of vertices of colors 1 and 2 has codegree 1 in H and the last vertex in thetriple has color 3. The latter event has probability 1 /
27, and therefore the expected numberof edges in H is | H | / ≥ | H | / (108 k ). Fix a coloring so that | H | ≥ | H | / (108 k ).Consider the bipartite graph G comprising all pairs of vertices of colors 1 and 2, so that | G | = | H | and G has average degree d G ≥ d / (108 k ). For convenience, let b > d G = 2 bn /k so | G | = bn /k . By Lemma 8, there exist constants δ, b > b > b , then G must contain at least δb k n copies of C k . Case 1. b ≥ /ǫ . By the pigeonhole principle, there exists an edge e such that the numberof C k containing e in G is at least2 kδb k n | G | ≥ kδb k − n − k . Let G ′ be the union of all 2 k -cycles in G containing e . Then by Lemma 9, for some constant c , | G ′ | ≥ cb k − n k = 12 cb k − d G ≥ k cǫ − − k − d > D provided n is large enough. Let C be a 2 k -cycle in G containing e . Then there exist edges e ∪ { v } , e ∪ { v } , . . . , e k ∪ { v k } in H where e , e , . . . , e k ∈ C and v , v , . . . , v k havecolor 3. Since H is B k -free, for some vertex z we have v = v = · · · = v k = z . Now thedegree of z in H is at least | G ′ | > D , which contradicts the fact that H has maximumdegree at most D . Case 2. b < /ǫ . In this case, d G < n /k /ǫ and so d < (108 k /ǫ ) n /k . By Lemma 5 on H , α ( H ) ≥ α ( H ) ≥ n d ≥ (cid:18) k ǫ (cid:19) − n k − k > n k − k − √ log n . This completes the proof. • Notice that Theorem 2 is valid for odd values of k , we believe that Theorem 1 shouldextend to odd values of k . An obstacle to applying the same idea as in the proof foreven values of k is that we don’t have “good” supersaturation for odd cycles. Newideas may be required to complete the proof for odd values.9 It seems likely that Theorem 1 can be extended to r -uniform hypergraphs with r ≥ r -uniform hypergraphs for r ≥ r -uniform version of Lemma 8). 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