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

On multicolor Ramsey numbers of triple system paths of length 3

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $\binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $\mathcal{L}$ be the loose 3-uniform path with 3 edges and $\mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $r_k(\mathcal{L}) < 1.55k$ and $r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large. The former result improves on the bound $r_k( \mathcal{L}) < 1.975k + 7\sqrt{k}$, which was recently established by {\L}uczak and Polcyn.Comment: 18 pages, 3 figure

Phase transitions in the Ramsey-Turán theory

Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√ n) = o (n2) for some constant c. We answer this question by proving the stronger RT(n, K5, o (√n log n)) = o(n2). It is known that RT (n, K5, c√n log n )= n2/4 + o (n2) for c > 1, so one can say that K5 has a Ramsey-Turán-phase transition at c√n log n. We extend this result to several other Kp's and functions f(n), determining many more phase transitions. We shall formulate several open problems, in particular, whether variants of the Bollobás-Erdos graph, which is a geometric construction, exist to give good lower bounds on RT (n, Kp, f(n)) for various pairs of p and f(n). These problems are studied in depth by Balogh-HuSimonovits, where among others, the Szemerédi's Regularity Lemma and the Hypergraph Dependent Random Choice Lemma are used.National Science Foundatio

Turán problems in graphs and hypergraphs

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. In Chapter 2, we prove an analogue of this result for 3-graphs (3-uniform hy¬pergraphs) together with an associated stability result. Let K− 4 , F5 and F6 be 3-graphs with vertex sets {1, 2,3, 4}, {1, 2,3,4, 5} and {1, 2,3,4, 5, 6} re¬spectively and edge sets E(K−4 ) = {123, 124, 134}, E(F5) = {123, 124, 345}, E(F6) = {123, 124,345, 156} and F = {K4, F6}. For n =6 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tri¬partite 3-graph S3(n). This extends an old result of Bollobas that S3(n) is the unique 3-graph of maximum size with no copy of K− 4 or F5. In 1941, Turán generalised Mantel's theorem to cliques of arbitrary size and then asked whether similar results could be obtained for cliques on hyper-graphs. This has become one of the central unsolved problems in the field of extremal combinatorics. In Chapter 3, we prove that the Turán density of K(3) 5 together with six other induced subgraphs is 3/4. This is analogous to a similar result obtained for K(3) 4 by Razborov. In Chapter 4, we consider various generalisations of the Turán density. For example, we prove that, if the density in C of ¯P3 is x and C is K3-free, then |E(C)| /(n ) ≤ 1/4+(1/4)J1 − (8/3)x. This is motivated by the observation 2 that the extremal graph for K3 is ¯P3-free, so that the upper bound is a natural extension of a stability result for K3. The question how many edges can be deleted from a blow-up of H before it is H-free subject to the constraint that the same proportion of edges are deleted from each connected pair of vertex sets has become known as the Turán density problem. In Chapter 5, using entropy compression supplemented with some analytic methods, we derive an upper bound of 1 − 1/('y(Δ(H) − /3)), where Δ(H) is the maximum degree of H, 3 ≤ 'y < 4 and /3 ≤ 1. The new bound asymptotically approaches the existing best upper bound despite being derived in a completely different way. The techniques used in these results, illustrating their breadth and connec¬tions between them, are set out in Chapter 1

Covering and tiling hypergraphs with tight cycles

Given $3 \leq k \leq s$, we say that a $k$-uniform hypergraph $C^k_s$ is a tight cycle on $s$ vertices if there is a cyclic ordering of the vertices of $C^k_s$ such that every $k$ consecutive vertices under this ordering form an edge. We prove that if $k \ge 3$ and $s \ge 2k^2$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + o(1))n$ has the property that every vertex is covered by a copy of $C^k_s$. Our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime. A perfect $C^k_s$-tiling is a spanning collection of vertex-disjoint copies of $C^k_s$. When $s$ is divisible by $k$, the problem of determining the minimum codegree that guarantees a perfect $C^k_s$-tiling was solved by a result of Mycroft. We prove that if $k \ge 3$ and $s \ge 5k^2$ is not divisible by $k$ and $s$ divides $n$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + 1/(2s) + o(1))n$ has a perfect $C^k_s$-tiling. Again our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime with $k$ even.Comment: Revised version, accepted for publication in Combin. Probab. Compu

Global hypercontractivity and its applications

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Tur\'an number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Tur\'an number, answering a question of Mubayi and Verstra\"ete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556

Dirac-type theorems in random hypergraphs

For positive integers $d<k$ and $n$ divisible by $k$, let $m_{d}(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_{1}(2,n)=\lceil n/2\rceil$. However, in general, our understanding of the values of $m_{d}(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In this paper we prove a "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, for any $d0$ and any "not too small" $p$, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and edge probability $p$ typically has the property that every spanning subgraph of $G$ with minimum degree at least $(1+\varepsilon)m_{d}(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_{d}(k,n)$ without actually knowing its value