Norm-graphs and bipartite tur�n numbers

Abstract

For every t > 1 and positive n we construct explicit examples of graphs G with IV(G)I =n, 2 ] IE(G)I > c t .n -• which do not contain a complete bipartite graph Kt,t!+l. This establishes the exact order of magnitude of the qSar~n numbers ex(n, Kt,s) for any fixed t and all s > t! + 1, improving over the previous probabilistic lower bounds for such pairs (t,s). The construction relies on elementary facts from commutative algebra.

Random algebraic construction of extremal graphs

“…It is for K 2,2 , and K 3,3 [6,5], but no constructions of K s,s -free graphs with Ω(n 2−1/s ) edges are known for any s ≥ 4. There are however constructions [12,2,4] of K s,t -free graphs with Ω(n 2−1/s ) edges when t is much larger than s. The aim of this paper is to present a new construction that uses both the algebra and probability. The construction is inspired by the construction in [4].…”

Turán Numbers of Subdivided Graphs

“…However, only for very few bipartite graphs F is the order of magnitude of ex(n, F ) even determined. Kövári, Sós, and Turán [16] showed that for fixed r, s, where 2 ≤ r ≤ s, ex(n, K r,s ) = O(n 2−1/r ) as a function of n. Kollár, Rónyai, and Szabó [15] showed that for fixed r, s, where r ≥ 4 and s ≥ r! + 1, ex(n, K r,s ) = Ω(n 2−1/r ) as a function of n, thus establishing the order of magnitude for such K r,s .…”

Constructive lower bounds for off-diagonal Ramsey numbers

“…By the well-known theorem of Erdős and Rado [8], there exists a ∆-system (=sunflower) with l petals (that is, l sets so that all intersections of a pair of them are identical) contained in X. Since the cliques in the sunflower are connected, there are In what follows we use the norm graphs of Kollár, Rónyai and Szabó [12]. (It is possible to get a slightly better bound using a modified version of these, described in [4], but this makes no essential difference for our purpose here.)…”

“…The proof is not long, but combines several tools from various mathematical areas. These include some ideas from algebraic geometry obtained in [12], the well known bound of Weil on character sums, spectral techniques and their connection to the pseudo-random properties of graphs, the known bounds of [13] for the problem of Zarankiewicz and the well known Erdős-Rado bound for the existence of ∆-systems.…”