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1. Off‐Diagonal Ramsey Numbers for Slowly Growing Hypergraphs

   https://doi.org/10.1002/rsa.21284  

   Mattheus, Sam; Mubayi, Dhruv; Nie, Jiaxi; Verstraëte, Jacques (January 2025, Random Structures & Algorithms)

   ABSTRACT For ak‐uniform hypergraph and a positive integer , the Ramsey number denotes the minimum such that every ‐vertex ‐free ‐uniform hypergraph contains an independent set of vertices. A hypergraph isslowly growingif there is an ordering of its edges such that for each . We prove that if is fixed and is any non‐k‐partite slowly growing ‐uniform hypergraph, then for ,In particular, we deduce that the off‐diagonal Ramsey number is of order , where is the triple system . This is the only 3‐uniform Berge triangle for which the polynomial power of its off‐diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs and hypergraph containers. 

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2. On the atomic structure of torsion-free monoids

   https://doi.org/10.1007/s00233-023-10385-8  

   Gotti, Felix; Vulakh, Joseph (October 2023, Semigroup Forum)

   Abstract LetMbe a cancellative and commutative (additive) monoid. The monoidMis atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also,Msatisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids. 

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3. Polynomial bounds for chromatic number VII. Disjoint holes

   https://doi.org/10.1002/jgt.22987  

   Chudnovsky, Maria; Scott, Alex; Seymour, Paul; Spirkl, Sophie (May 2023, Journal of Graph Theory)

   Abstract Aholein a graph is an induced cycle of length at least four, and a ‐multiholein is the union of pairwise disjoint and nonneighbouring holes. It is well known that if does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any integer , if does not contain a ‐multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially ‐bounded. 

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4. Every positive integer is the order of an ordinary abelian variety over $${{\mathbb {F}}}_2$$

   https://doi.org/10.1007/s40993-021-00274-w  

   Howe, Everett W.; Kedlaya, Kiran S. (September 2021, Research in Number Theory)

   Abstract We show that for every integer$$m > 0$$ $m>0$, there is an ordinary abelian variety over $${{\mathbb {F}}}_2$$ $F_2$that has exactlymrational points. 

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5. Rainbow even cycles

   Dong, Zichao; Xu, Zijian (April 2024, SIAM journal on discrete mathematics)

   We prove that every family of (not necessarily distinct) even cycles D_1,...,D_{1.2n-1} on some fixed n-vertex set has a rainbow even cycle (that is, a set of edges from distinct D_i’s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, the result is best possible for every positive integer n. 

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