On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

Abstract

Abstract. The Erdős-Rényi and Projective Norm graphs are algebraically defined graphs that have proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their eigenvalues have been computed and this yields an upper bound on their independence number. Here we show that in many cases, this upper bound is sharp in order of magnitude.Our result for the Erdős-Rényi graph has the following reformulation: the maximum size of a family of mutually non-orthogonal lines in a vector space of dimension three over the finite field of order q is of order q 3/2 .We also prove that every subset of vertices of size greater than q 2 /2 + q 3/2 + O(q) in the Erdős-Rényi graph contains a triangle. This shows that an old construction of Parsons is asymptotically sharp. Several related results and open problems are provided.

New Bounds on Even Cycle Creating Hamiltonian Paths Using Expander Graphs

“…For the problem of determining RP (n), Conjecture 5.1 would imply such a large gap: Cibulka [5] proved that n Let p be an odd prime, and let F p denote the finite field of p elements. Let the vertex set of the graphG =G(p) be the vector space F 2 p , and let two vertices (a, c), (b, d) ∈ F 2 p be adjacent inG if and only if ab = c + d. This is a well-known construction for a dense C 4 -free graph, and the spectral properties ofG have already been studied, see for example the work of Mubayi-Williford [28] and Solymosi [31]. Thus it is already established in the literature thatG satisfies all our requirements except that it contains loops.…”

On the independence number of graphs related to a polarity

“…The next result summarizes lower and upper bounds on the size of the independence number of ER q studied in [12,15].…”

Some constructive bounds on Ramsey numbers

“…P π q has n = q 2 + q + 1 vertices; q + 1 vertices have degree q, and q 2 vertices have degree q + 1, hence the number of edges is 1 2 q(q + 1) 2 . This graph is also known as the Erdős-Rényi graph; for more information on its independence number, see [9]. The second largest absolute value of an eigenvalue of the polarity graph with loops P π ,o q is λ = √ q.…”