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Let H 1 , ..., H k be bipartite graphs. The bipartite Ramsey number b r (H 1 , ... , H k) is the minimum integer N such that any k -edge-coloring of the complete bipartite graph K N , N contains a monochromatic H i in color i for some i ∈ { 1 , 2 , ... , k }. The study of bipartite Ramsey number was initiated in the early 70s by Faudree and Schelp (1975), and they showed that b r (P 2 n , P 2 m) = n + m − 1 , where P n denotes a path with n vertices. Combining their result and the regularity lemma, one can obtain asymptotic value b r (C 2 ⌊ α 1 n ⌋ , C 2 ⌊ α 2 n ⌋ ) = (α 1 + α 2 + o (1)) n for α 1 , α 2 > 0 , where C n denotes a cycle with n vertices. The study of bipartite Ramsey numbers of cycles has also attracted lots of attention recently. For example, Bucić, Letzter and Sudakov (2019) showed that b r (C 2 n , C 2 n , C 2 n) = (3 + o (1)) n and they remarked that it is interesting to determine the exact value of b r (C 2 n , C 2 m). Zhang and Sun (2011), Zhang, Sun, and Wu (2013), and Gholami and Rowshan (2021) determined b r (C 4 , C 2 n) , b r (C 6 , C 2 n) , and b r (C 8 , C 2 n) respectively. In this paper, we determine b r (C 2 n , C 2 m) for all the remaining cases and show that b r (C 2 n , C 2 m) = n + m − 1 for all n > m ≥ 5 and b r (C 2 n , C 2 n) = 2 n if n ≥ 5. It is somehow interesting that b r (C 2 n , C 2 m) is the same as b r (P 2 n , P 2 m) if n ≠ m , and b r (C 2 n , C 2 n) is one more than b r (P 2 n , P 2 n). [ABSTRACT FROM AUTHOR]

The celebrated Erdős–Ko–Rado theorem (Erdős et al., 1961) states that a maximum intersecting k -uniform family on [ n ] must be a full star if n ≥ 2 k + 1. Furthermore, Hilton and Milner (1967) showed that if an intersecting k -uniform family on [ n ] is not a subfamily of a full star, then its maximum size is achieved only by a family isomorphic to H M (n , k) ≔ { G ∈ [ n ] k : 1 ∈ G , G ∩ [ 2 , k + 1 ] ≠ 0̸ } ∪ { [ 2 , k + 1 ] } if n ≥ 2 k + 1 and k ≥ 4 , and there is one more possibility if k = 3. Han and Kohayakawa (2017) determined a maximum intersecting k -uniform family on [ n ] which is neither a subfamily of a full star nor a subfamily of the extremal families in Hilton-Milner theorem, and they asked what the next maximum intersecting k -uniform families on [ n ] are. Kostochka and Mubayi (2017) answered the question for large enough n. In this paper, we are going to get rid of the requirement that n is large enough in the result by Kostochka and Mubayi (2017), and answer the question of Han and Kohayakawa (2017) for all n ≥ 2 k + 1. [ABSTRACT FROM AUTHOR]

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. The Lagrangian density of an r -uniform graph F is π λ ( F ) = sup { r ! λ ( G ) : G i s F - f r e e } , where λ ( G ) is the Lagrangian of an r -uniform graph G . Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. In particular, Hefetz and Keevash (2013) studied the Lagrangian density of the 3-uniform matching of size 2 and the Turán number of its extension. We obtain the Lagrangian densities of a 3-uniform matching of size t , a 3-uniform linear star of size t , and a 4-uniform linear star of size t . Using a stability argument of Pikhurko and a transference technique between the Lagrangian density of an r -uniform graph F and the Turán number of its extension, we can also determine the Turán numbers of their extensions. [ABSTRACT FROM AUTHOR]

Abstract: Text: A positive integer n is perfect if the sum of the proper positive divisors of n equals n. Generalizing this we call n a Zumkeller number if the set of its positive divisors can be partitioned into two disjoint subsets of equal sum. Similarly we call n a half-Zumkeller number if the set of its proper positive divisors can be so partitioned. A study of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. Clark et al. (2008) [1] announced some results about Zumkeller numbers and half-Zumkeller numbers, and suggested two conjectures. In the present paper we shall settle one of the conjectures, prove the second conjecture in some special cases, and prove several results related to the second conjecture. We shall also show that if there is an even Zumkeller number that is not half-Zumkeller it is bigger than . Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=z85qyvIorBE. [Copyright &y& Elsevier]

Abstract: Let be an integer. The real number is a jump for r if there exists such that for every positive ϵ and every integer , every r-uniform graph with vertices and at least edges contains a subgraph with m vertices and at least edges. A result of Erdős, Stone and Simonovits implies that every is a jump for . For , Erdős asked whether the same is true and showed that every is a jump. Frankl and Rödl gave a negative answer by showing that is not a jump for r if and . Another well-known question of Erdős is whether is a jump for and what is the smallest non-jumping number. In this paper we prove that is not a jump for . We also describe an infinite sequence of non-jumping numbers for . [Copyright &y& Elsevier]

A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if G is a graph on n vertices with at least ⌊ n 2 / 4 ⌋ + 1 edges, then G contains at least ⌊ n / 2 ⌋ triangles. We prove a spectral version of Erdős–Rademacher's theorem. Moreover, Mubayi (2010) [28] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs. [ABSTRACT FROM AUTHOR]