Ramsey-Turán numbers for semi-algebraic graphs

A semi-algebraic graph G = (V,E) is a graph where the vertices are points in Rd, and the edge set E is defined by a semi-algebraic relation of constant complexity on V . In this note, we establish the following Ramsey-Turán theorem: for every integer p > 3, every Kp-free semi-algebraic graph on n vertices with independence number o(n) has at most 12 ( 1− 1 dp/2e−1 + o(1) ) n2 edges. Here, the dependence on the complexity of the semi-algebraic relation is hidden in the o(1) term. Moreover, we show that this bound is tight. Mathematics Subject Classifications: 05D10, 52C10


Introduction
Over the past decade, several authors have shown that many classical theorems in extremal graph theory can be significantly improved if we restrict our attention to semi-algebraic graphs, that is, graphs whose vertices are points in Euclidean space, and edges are defined by a semi-algebraic relation of constant complexity [1,5,8,11,9,4]. In this note, we continue this sequence of works by studying Ramsey-Turán numbers for semi-algebraic graphs.
More formally, a graph G = (V, E) is a semi-algebraic graph with complexity at most t, if its vertex set V is an ordered set of points in R d , where d ≤ t, and if there are at most t polynomials g 1 , . . . , g s ∈ R[x 1 , . . . , x 2d ], s ≤ t, of degree at most t and a Boolean formula Φ such that for vertices u, v ∈ V such that u comes before v in the ordering, At the evaluation of g (u, v), we substitute the variables x 1 , . . . , x d with the coordinates of u, and the variables x d+1 , . . . , x 2d with the coordinates of v. Here, we assume that the complexity t is a fixed parameter, and n = |V | tends to infinity. The classical theorem of Turán gives the maximum number of edges in a K p -free graph on n vertices. Theorem 1.1 (Turán, [13]). Let G = (V, E) be a K p -free graph with n vertices. Then The only graph for which this bound is tight is the complete (p − 1)-partite graph whose parts are of size as equal as possible. This graph can easily be realized as an intersection graph of segments in the plane, which is a semi-algebraic graph with complexity at most four. Therefore, Turán's theorem cannot be improved by restricting it to semi-algebraic graphs. Let H be a fixed graph. The Ramsey-Turán number RT(n, H, α) is defined as the maximum number of edges that an n-vertex graph of independence number at most α can have without containing H as a (not necessarily induced) subgraph. Ramsey-Turán numbers were introduced by Andrásfai [2] and were motivated by the classical theorems of Ramsey and Turán and their connections to geometry, analysis, and number theory. According to one of the earliest results in Ramsey-Turán theory, which appeared in [7], for every p ≥ 2, we have For excluded K 4 , a celebrated result of Szemerédi [12] and Bollobás-Erdős [3] states that This was generalized by Erdős, Hajnal, Sós, and Szemerédi [6] to all cliques of even size. For every p ≥ 2, we have For more results in Ramsey-Turán theory, consult the survey of Simonovits and Sós [10].
In the present note, we establish asymptotically tight bounds on Ramsey-Turán numbers for semi-algebraic graphs. We define RT t (n, K p , o(n)) as the maximum number of edges that n-vertex K p -free semi-algebraic graphs with complexity at most t can have, if their independence number is o(n). Strictly speaking, this definition and all above results apply to sequences of graphs with n vertices, as n tends to infinity.
It turns out that if the size of the excluded clique is even, then the answer to the Ramsey-Turán question significantly changes when the graphs are required to be semi-algebraic. However, in the odd case, we obtain the same asymptotics for the Ramsey-Turán function as in (1). More precisely, we have Theorem 1.2. For any fixed integers t ≥ 10 and p ≥ 2, we have 2 Proof of Theorem 1.2 The aim of this section is to prove Theorem 1.2. One of the main tools used in the proof is the following regularity lemma for semi-algebraic graphs. Given a graph G = (V, E), a vertex partition is called equitable if any two parts differ in size by at most one. Given two disjoint subsets Then V has an equitable partition V = V 1 ∪ · · · ∪ V K into K part, where 1/ε < K < (1/ε) c , such that all but an ε-fraction of the pairs of parts are homogeneous.

The upper bound in Theorem 1.2 follows from
Theorem 2.2. Let ε > 0 and let G = (V, E) be an n-vertex semi-algebraic graph with complexity at most t. If G is K 2p -free and |E| > 1 2 1 − 1 p−1 + ε n 2 , then G has an independent set of size γn, where γ = γ(t, p, ε).
Proof. We apply Lemma 2.1 with parameter ε/4 to obtain an equitable partition P : where c = c(t) and all but an at most ε 4 -fraction of all pairs of parts in P are homogeneous (complete or empty with respect to E). If n ≤ 10K, then G has an independent set of size one, and the theorem holds trivially. So, we may assume n > 10K.
By deleting all edges inside each part, we have deleted at most ≤ ε n 2 5 edges. Deleting all edges between non-homogeneous pairs of parts, we lose an additional at most n K 2 ε 4 K 2 2 ≤ ε n 2 5 edges. In total, we have deleted at most 2εn 2 /5 edges of G. The only edges that remain in G are edges between homogeneous pairs of parts, and we have at least 1 2 1 − 1 p−1 + ε/5 n 2 edges. By Turán's theorem (Theorem 1.1), there is at least one remaining copy of K p , and its vertices lie in p distinct parts V i 1 , . . . , V ip ∈ P that form a complete p-partite subgraph. If any of the parts V i j forms an independent set in G, then there is an independent set of order |V i j | ≥ n/K ≥ γn, where γ = γ(t, , p), and we are done. Otherwise, there is an edge in each of the p parts, and the endpoints of these p edges form a K 2p in G, a contradiction.
The lower bound on RT(n, K 2p−1 , o(n)) and RT(n, K 2p , o(n)) in Theorem 1.2 is constructive and is based on the following result of Walczak.

Lemma 2.3 ([14]
). For any pair of positive integers n and p, where n is a multiple of p − 1, there is a collection S of n/(p − 1) segments in the plane whose intersection graph G S is triangle-free and has no independent set of size c p n/ log log n. Here c p is a suitable constant.
The construction. Take p − 1 dilated copies of a set S meeting the requirements in Lemma 2.3, and label them as S 1 , . . . , S p−1 , so that S i lies inside a ball with center (i, 0) and radius 1/10. Set V = S 1 ∪ · · · ∪ S p−1 . Note that |S i | = n/(p − 1) so that |V | = n. Let G = (V, E) be the graph whose vertices are the elements of V , and two vertices (that is, two segments) are connected by an edge if and only if they cross or their left endpoints are at least 1/2 apart. The graph G consists of a complete (p − 1)-partite graph, where each part induces a copy of the triangle-free graph G S . Clearly, G is K 2p−1 -free and does not contain any independent set of size c p n/ log log n. Moreover, Since every segment can be represented by a point in R 4 and the intersection and distance relations have bounded description complexity (see [1]), E is a semi-algebraic relation with complexity at most c, where c is an absolute constant.