Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2022-01-04 , DOI: 10.1016/j.jctb.2021.12.004 M.C. Laskowski , C.A. Terry
Given a finite relational language , a hereditary -property is a class of finite -structures closed under isomorphism and substructure. The speed of is the function which sends an integer to the number of distinct elements in with underlying set . In this paper we give a description of many new jumps in the possible speeds of a hereditary -property, where is any finite relational language. In particular, we characterize the jumps in the polynomial and factorial ranges, and show they are essentially the same as in the case of graphs. The results in the factorial range are new for all examples requiring a language of arity greater than two, including the setting of hereditary properties of k-uniform hypergraphs for . Further, adapting an example of Balogh, Bollobás, and Weinreich, we show that for all , there are hereditary properties of k-uniform hypergraphs whose speeds oscillate between functions near the upper and lower bounds of the penultimate range, ruling out many natural functions as jumps in that range. Our theorems about the factorial range use model theoretic tools related to the notion of mutual algebraicity.
中文翻译:
有限关系语言中遗传属性的速度跳跃
给定一个有限关系语言 ,世袭的 - 属性是一个类 有限的 -在同构和子结构下封闭的结构。的速度 是发送整数的函数 到不同元素的数量 有基础集合 . 在这篇论文中,我们描述了遗传的可能速度的许多新跳跃。-财产,在哪里 是任何有限关系语言。特别是,我们描述了多项式和阶乘范围内的跳跃,并表明它们与图的情况基本相同。对于所有需要元数大于 2 的语言的示例,阶乘范围中的结果都是新的,包括设置k均匀超图的遗传特性. 此外,通过改编 Balogh、Bollobás 和 Weinreich 的示例,我们表明对于所有,有k均匀超图的遗传特性,其速度在倒数第二个范围的上限和下限附近的函数之间振荡,排除了许多自然函数作为该范围内的跳跃。我们关于阶乘极差的定理使用了与互代数概念相关的模型理论工具。