Recall that in extremal graph theory, we would like to answer questions of the following sort: ‘What is the maximum/minimum possible parameter C among graphs satisfying a certain property P?’ In the last lecture, we see Mantel’s theorem, which answers the above question with the parameter being the number of edges and the property being triangle-free. In this lecture, we will be looking at othe...

This paper is a survey on the topic in extremal graph theory influenced directly or indirectly by Paul Turán. While trying to cover a fairly wide area, I will try to avoid most of the technical details. Areas covered by detailed fairly recent surveys will also be treated only briefly. The last part of the survey deals with random ±1 matrices, connected to some early results of Szekeres and Turán.

Let f(v) denote the maximum number of edges in a graph of order v and of girth at least 5. In this paper, we discuss algorithms for constructing such extremal graphs. This gives constructive lower bounds of f(v) for v ≤ 200. We also provide the exact values of f(v) for v ≤ 24, and enumerate the extremal graphs for v ≤ 10.

Let G be a simple graph on the vertex set [n] with edge set E(G) and let S be the polynomial ring K[x1, . . . , xn, y1, . . . , yn] in 2n variables endowed with the lexicographic order induced by x1 > · · · > xn > y1 > · · · > yn. The binomial edge ideal JG ⊂ S associated with G is generated by all the binomials fij = xiyj−xjyi with {i, j} ∈ E(G). The binomial edge ideals were introduced in [5]...

G(?z; I) will denote a graph of n vertices and 1 edges. Let fO(lz, K) be the smallest integer such that there is a G (n; f,, (n, k)) in which for every set of K vertices there is a vertex joined to each of these. Thus for example fO(3, 2) = 3 since in a triangle each pair of vertices is joined to a third. It can readily be checked that f,(4, 2) = 5 (the extremal graph consists of a complete 4-g...

The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The (p, q)-extremal problem consists in finding the maximum number of edges on a k-uniform hypergraph H with n vertices such that among any p edges some q of them have...

The harmonic index H(G) of a graph G is the sum of 2 d(u) + d(υ) over all edges uυ of G, where d(u) denotes the degree of a vertex u in G. In this paper, we give the minimum value of H(G) for graphs G with given minimum degree δ(G) ≥ 2 and characterize the corresponding extremal graph. Furthermore, we prove a best-possible lower bound on the harmonic index of a triangle-free graph G with arbitr...

The harmonic index of a graph G is defined as the sum of the weights 2 d(u) + d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this work, we present the minimum and maximum values of the harmonic index for connected graphs with girth at least k (k ≥ 3), and characterize the corresponding extremal graphs. Using this result, we obtain several relations between the h...

Theorem 1.3 (RegLem) For every ǫ > 0 and positive integer m, there exist two integers M(ǫ,m) and N(ǫ,m) with the property that, for every graph G with n ≥ N(ǫ,m) vertices, there exists a partition of the vertex set into k + 1 classes V = V0 + V1 + · · ·+ Vl such that • m ≤ l ≤ M(ǫ,m) • |V0| < ǫn • |V1| = |V2| = · · · = |Vl| • For distinct i, j 6= 0, all but at most ǫl2 of the pairs (Vi, Vj) are...

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, an...