Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short self-contained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by Furstenberg-Katznelson (1978). 1.

We prove a removal lemma for infinitely-many forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constant-time probabilistic algorithm to edit a small number of edges. It also gives a quantitative bound in terms of a coloring number of the property. It is based on a new hypergraph regularity lemma [14].

Let r ≥ 3 and (c/r r) r lnn ≥ 1. If G is a graph of order n and its largest eigenvalue µ (G) satisfies µ (G) ≥ 1 − 1 + c n, r − 1 then G contains a complete r-partite subgraph with r − 1 parts of size ⌊(c/r r) r ln n ⌋ and one part of size greater than n1−cr−1. This result implies the Erdős-Stone-Bollobás theorem, the essential quantitative form of the Erdős-Stone theorem. Moreover, if F is a fixed graph with chromatic number r, then 1 1 lim max {µ (G) : G is of order n and F � G} = 1 − n→ ∞ n r − 1. This result implies the Erdős-Stone-Simonovits theorem. Keywords: largest eigenvalue; r-partite subgraph; Erdős-Stone-Bollobás theorem; Erdős-Stone-Simonovits theorem This note is part of an ongoing project aiming to build extremal graph theory on spectral grounds, see, e.g., [6], [14, 22]. The fundamental Erdős-Stone theorem [9] states that, given r ≥ 3 and c> 0, every graph with n vertices and ⌈(1 − 1 / (r − 1) + c)n 2 /2 ⌉ edges contains a complete r-partite graph with each part of size g (n, r, c) , where g (n, r, c) tends to infinity with n. In [4] Bollobás and Erdős found that g (r, c, n) = Θ (log n) , and in [3], [5], [8], and [12] the function g (r, c, n) was determined with great precision. Here we deduce the Erdős-Stone-Bollobás result from a weaker, spectral condition. Our notation follows [2]. Let Kr (s1,..., sr) be the complete r-partite graph with parts of size s1,...,sr, and let µ (G) be the largest adjacency eigenvalue of a graph G. Our main result: Theorem 1 Let r ≥ 3, (c/rr) r ln n ≥ 1, and G be a graph with n vertices. If µ (G) ≥ 1 − 1 + c n, r − 1 then G contains a Kr (s,...s, t) with s ≥ ⌊(c/rr) r ln n ⌋ and t> n1−cr−1. 1 As an easy consequence, we strengthen the Erdős-Stone-Simonovits theorem [10]. Theorem 2 Let r ≥ 3 and F be a fixed graph with chromatic number r. Then

Abstract. We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. showed in 1983. While Cooley et al. showed it for the 2-color case recently and independently from the author, our theorem contains the multicolor case and our proof is simple and provides a stronger embedding lemma without depending on heavy theorems by Rödl-Schacht [18] that their proof is based on. 1.

Graphs with many r-cliques have large complete

Abstract. We show that the the Ramsey number of every bounded-degree uniform hypergraph is linear with respect to the number of vertices. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [8] showed in 1983. Our result may demonstrate the potential of a new hypergraph regularity lemma by [18]. 1.

Graphs with many r-cliques have large complete r-partite subgraphs

Let r ≥ 2 and c> 0. If G is a graph of order n and the largest eigenvalue of its adjacency matrix satisfies µ (G) ≥ (1 − 1/r ⌊ + c) n, then G contains a complete r-partite subgraph with r − 1 parts of size (c/(r + 1) r) r+1 ⌋ lnn and one part of size greater than n1−cr. This result implies a quantitative form of the Erdős-Stone theorem. Moreover, if F is a fixed graph with chromatic number r + 1, then 1 lim n→ ∞ n r − 1 max {µ (G) : G is of order n and F � G} =. r This result implies the Erdős-Stone-Simonovits theorem. Keywords: largest eigenvalue; r-partite subgraph; Erdős-Stone theorem; Erdős-Stone-Simonovits theorem The fundamental Erdős-Stone theorem [7] states that, given r ≥ 2 and c> 0, every graph with n vertices and ⌈(1 − 1/r + c) n 2 /2 ⌉ edges contains a complete (r + 1)-partite graph with each part of size g (n, r, c) , where g (n, r, c) tends to infinity with n. In [3], [2], [4], [6], and [9] the function g (n, r, c) was determined with great precision. Put simply: for r and c fixed, g (n, r, c) = Ω (log n). The main aim of this note is to deduce the same conclusion from a weaker spectral condition. Our notation follows [1]; thus, Kr (s1,...,sr) denotes a complete r-partite graph with parts of size s1,...,sr. Specifically, we write kr (G) for the number of r-cliques of a graph G and µ (G) for the largest eigenvalue µ (G) of its adjacency matrix. Here is our main result. Theorem 1 Let r ≥ 2, c> 0, and G be a graph with n vertices. If µ (G) ≥ (1 − 1/r + c)n, (1) then G contains a Kr (s,...s, t) with s ≥ (c / (r + 1) r) r+1 ⌋ ln n and t> n1−cr. 1 In view of µ (G) ≥ 2e (G)/v (G) , Theorem 1 implies a quantitative form of the Erdős-Stone theorem: if e (G) ≥ (1 − 1/r + c) n2 /2, then G contains a Kr+1 (s,..., s, t) with s ≥ (c / (r + 1) r) r+1 ⌋ ln n, t> n 1−cr This bound is stronger than the one given in [2] and comparable with those established