A Construction for Clique-Free Pseudorandom Graphs

Abstract

A construction of Alon and Krivelevich gives highly pseudorandom K k -free graphs on n vertices with edge density equal to Θ(n −1/(k−2) ). In this short note we improve their result by constructing an infinite family of highly pseudorandom K k -free graphs with a higher edge density of Θ(n

Multicolor Ramsey Numbers via Pseudorandom Graphs

“…Because of this, the existence of weakly optimal K s -free (n, d, λ)-graphs is indeed weaker than the existence of optimal ones. Sudakov, Szabó, and Vu [11] conjectured the existence of optimal K s -free (n, d, λ)graphs for all s 3 and all n; such graphs where constructed by Alon [2] in the case s = 3 but the conjecture remains open for s 4 (see [6] for the best known construction for s 5, which agrees with Alon's bound for s = 4). Conditional on this conjecture, Mubayi and Verstraëte showed that r(s, t) grows like t s−1 up to polylogarithmic factors.…”

A clique-free pseudorandom subgraph of the pseudo polarity graph

“…It would be interesting to provide geometrical constructions leading to even denser pseudorandom graphs, ideally breaking the barrier of Θ(n −1/k ). The idea of taking large subgraphs of polarity graphs applied here and in [BIP20] seems to reach its limits, at least for case of k = 3 on which the induction relies: a K 3 -free graph obtained from any polarity graph G q (3, ⊥) has at most α(G q (3, ⊥)) = Θ(q 3/2 ) vertices as shown by Mubayi and Williford [MW07].…”

“…These three reasons, along with the construction of the subgraph of non-absolute points are also contained in [AK97]. The main insight of [BIP20] is that it is possible, when q is odd and ⊥ orthogonal, to take a large subgraph on non-absolute points which reduces the clique number even further. In this way, they find a subgraph of the same density d/n = Θ(n −1/(k−1) ), but which is K k -free.…”

“…In [AK97], Alon and Krivelevich obtained optimally pseudorandom K k -free graphs with d n = Ω n −1/(k−2) , by considering a symmetric non-alternating bilinear form of F k−1 q , q even. Recently, Bishnoi, Ihringer and Pepe [BIP20], by using a symmetric bilinear form of F k q , q odd, provided a construction of optimally pseudorandom K k -free graphs with d n = Ω n −1/(k−1) and this is currently the best known construction in terms of edge density for optimally pseudorandom K k -free graphs, k ≥ 4. Here we give a new family of optimally pseudorandom K k -free graphs with d n = Ω n −1/(k−1) , arising from a symmetric non-alternating bilinear form of F k q , q even.…”

A note on pseudorandom Ramsey graphs

“…Alon and Krivelevich [4] gave a construction of K s -free (n, d, λ)-graphs with d = Ω(n 1−1/(s−2) ) and λ = O( √ d) for all s ≥ 3, and this was slightly improved by Bishnoi, Ihringer and Pepe [7] to obtain d = Ω(n 1−1/(s−1) ). This is the current record for the degree of a K s -free (n, d, λ)-graph…”