High dimensional Hoffman bound and applications in extremal combinatorics

One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Tur\'an problem for (hyper-)graphs, can be encoded as special cases of this problem. We also give a new generalization of the Hoffman bound for hypergraphs which is sharp for the tensor power of a hypergraph whenever it is sharp for the original hypergraph. As an application of our Hoffman bound, we make progress on the problem of Frankl on families of sets without extended triangles from 1990. We show that if $\frac{1}{2}n\le2k\le\frac{2}{3}n,$ then the extremal family is the star, i.e. the family of all sets that contains a given element. This covers the entire range in which the star is extremal. As another application, we provide spectral proofs for Mantel's theorem on triangle-free graphs and for Frankl-Tokushige theorem on $k$-wise intersecting families

Convex Analysis And Flows In Infinite Networks

We study the existence of flows in infinite networks and extend basic theorems due to Gale and Hoffman and to Ford and Fulkerson. The classical approach to finite networks uses a constructive combinatorical algorithm that has become known as the labelling algorithm. Our approach to infinite networks involves Hahn--Banach type theorems on the existence of certain linear functionals. Thus the main tools are from the theory of functional and convex analysis. In Chapter II, we discuss sublinear and linear functionals on real vector spaces in the spirit of the work of K {o}nig. In particular, a generalization of K {o}nig\u27s minimum theorem is established. Our theory leads to some useful interpolation results. We also establish a variant of the main interpolation theorem in the context of convex cones. We reformulate the results of Ford--Fulkerson and Gale--Hoffman in terms of certain additive and biadditive set functions. In Chapter III, we show that the space of all additive set functions may be canonically identified with the dual space of a space of certain step functions and that the space of all biadditive set functions may be identified with the dual space of a space of certain step functions in two variables. Our work an additive set functions is in the spirit of classical measure theory, while the case of biadditive set functions resembles the theory of product measures. In Chapter IV, we develop an extended version of the Gale--Hoffman theorem on the existence of flows in infinite networks in a setting of measure-theoretic flavor. This general flow theorem is one of our central results. We discuss, as an application of our flow theorem, a Ford--Fulkerson type result on maximal flows and minimal cuts in infinite networks containing sources and sinks. In addition, we present applications to flows in locally finite networks and to the existence of antisymmetric flows under certain natural conditions. We conclude with a discussion of the case of triadditive set functions. In the appendix, we review briefly the classical theory of maximal flows and minimal cuts in networks with finitely many nodes

Quantitative Tverberg theorems over lattices and other discrete sets

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carath\'eodory's theorems.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1503.0611

Hyperplane families creating envelopes

Let $N$ be an $n$-dimensional $C^\infty$ manifold and let $\widetilde{\varphi}: N\to \mathbb{R}^{n+1}$, $\widetilde{\nu}: N\to S^n$ be $C^\infty$ mappings. We first give a necessary and sufficient condition for the hyperplane family $\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})}$ defined by $\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})} =\cup_{x\in N}\left\{X\in \mathbb{R}^{n+1}\; |\; \left(X-\widetilde{\varphi}(x)\right)\cdot \widetilde{\nu}(x)=0\right\}$ to create an envelope (Theorem 1). As a by-product of the proof of Theorem 1, when the given hyperplane family $\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})}$ creates an envelope $\widetilde{f}: N\to \mathbb{R}^{n+1}$, an explicit expression of the envelope $\widetilde{f}$ is obtained in terms of $\widetilde{\varphi}$ and $\widetilde{\nu}$ (Corollary 2). The vector formula given in Corollary 2 holds even at a singular point of $\widetilde{\nu}$ so long as the hyperplane family $\mathcal{H}_{(\widetilde{\varphi}, \widetilde{\nu})}$ creates an envelope. In this sense, Corollary 2 may be regarded as a complete generalization of the celebrated Cahn-Hoffman vector formula. Moreover, we give a criterion when and only when $\mathcal{H}_{\left(\widetilde{\varphi}, \widetilde{\nu}\right)}$ creates a unique envelope (Theorem 2).Comment: Added a large amount of expository documents. Added many figures. Added examples. 26 pages, 17 figure

Quantitative Tverberg, Helly, & Carath\'eodory theorems

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.Comment: 33 page

A characterization and an application of weight-regular partitions of graphs

A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $\nu_u$ of the Perron eigenvector $\mathbf{\nu}$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity by using a Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular graphs. In addition, we show an application of weight-regular partitions to study graphs that attain equality in the classical Hoffman's lower bound for the chromatic number of a graph, and we show that weight-regularity provides a condition under which Hoffman's bound can be improved