The Bipartite Ramsey Number Of Cycles And Turán Density Of Hypergraphs

Ramsey Theory and Turán problem are central to the development of ex-tremal combinatorics.Let r,k ≥2 be integers and H1,...,Hk be given r-uniform hypergraphs.The Ramsey number Rr(H1,...,Hk)is the minimum integer N such that any k-edge coloring of the complete r-uniform hypergraph KN(r)contains a monochromatic subgraph in color i isomorphic to Hi for some 1 ≤i ≤k.If H1,H2,...,Hk are all the complete r-uniform hypergraphs,then Rr(H1,H2,...,Hk)is the classical Ramsey number.At present,the known classical Ramsey numbers and their upper and lower bounds are still very few.Solving the exact value of various Ramsey number and estimating their upper and lower bounds have been a hot and difficult topic in the study of Ramsey Theory.Given a positive integer n and an r-uniform hypergraph F,the Turán number ex(n,F)is the maximum number of edges in an F-free r-uniform hypergraph on n vertices.The Turán density of F is defined as π(F)=lim ex(rn)(n,F).How to determine Turán numbers and Turán densities of hypergraphs?This is a challenging extremal problem in combinatorial mathematics.Especially for r-uniform hypergraphs,where r≥ 3,there are very few results on this problem.Regularity lemma is an important tool in the study of extremal combinatorics.In 1973,Bondy-Erdos conjectured that if k≥2 and n≥ 3 is odd,then R(k;Cn)=2k-1(n-1)+1.Benevides and Skokan proved that there exists n1 such that for even n≥,R(3;Cn)=2n.Ferguson determined the exact value of the Ramsey number for a triple of two long even cycles and one long odd cycle.Figaj and Luczak determined the asymptotic value of the Ramsey number for a triple of long cycles.They all applied the Regularity lemma.Let B1,...,Br be bipartite graphs and an integer r≥ 2.The bipartite Ramsey number ber br(B1,...,Br)is the minimum integer N such that any r-edge coloring of the complete bipartite graph KN,N contains a monochromatic subgraph in color i isomorphic to Bi for some 1 ≤i ≤r.The study of the bipartite Ramsey number was initiated in the early 70s by Faudree and Schelp.Later,the research on the bipartite Ramsey number aroused the interest of researchers.A natural idea is to extend the Ramsey number of cycles to the case of the bipartite Ramsey numbers.Luo-Peng determined the asymptotic value of the bipartite Ramsey numbers of a triple of long cycles.Joubert and Hattingh-Joubert gave a weaker upper bound of multicolored bipartite Ramsey numbers of short cycles respectively.In this paper,we obtain an upper bound of multicolored bipartite Ramsey number of cycle C2n for n large enough and the asymptotic value of multicolored bipartite Ramsey number of large cycles under some conditions by applying the Regularity lemma.Motivated by a new class of Ramsey-Turán type problems raised by Schelp.The intent is to find the smallest positive constant 0<c<1 such that if δ(G)>≥c|V(G)| and |V(G)|≥R(r;H),then any r-edge-coloring of G contains a monochromatic subgraph H.In this paper,we also obtain a weaker bipartite Ramsey-Turán type result by applying the degree form of the Regularity lemma.Lagrangian is an important tool in the study of the Turán problems.There is a close relationship between jump numbers and Turin densities.A numberα ∈[0,1)is a jump for r≥ 2 if and only if there exists a constant ∈>0 such thatΓ∪(α,α+∈)=(?),where Γr is the set of all possible Turán densities of r-uniform graphs.Erdos proposed the well-known jumping constant conjecture:Every α ∈[0,1)is a jump for every integer r≥ 2.The conjecture holds true for general graphs(2 graphs).For hypergraphs,Frankl and Rodl obtained a sequence of non-jumping numbers by applying the Lagrangian method and disproved this conjecture.Using a similar approach,more non-jumping numbers were obtained.In this paper,we construct a set of infinite non-jump numbers by applying the Lagrangian method or new non-jump numbers by using the known non-jump numbers.In this paper,we also apply the Lagrangian method to determine the Turán density of the extension of the(r-3)-fold enlargement of a 3-uniform matching.