Asymptotic Improvements to the Lower Bound of Certain Bipartite Turán Numbers

Abstract

Abstract. We show that there are graphs with n vertices containing no K 5,5 which have about 1 2 n 7/4 edges, thus proving that ex(n, K 5,5 ) ≥ 1 2 (1 + o(1))n 7/4 . This bound gives an asymptotic improvement to the known lower bounds on ex(n, K t,s ) for t = 5 when 5 ≤ s ≤ 12, and t = 6 when 6 ≤ s ≤ 8.

Forbidden subgraphs in the norm graph

“…In [1] Alon, Rónyai and Szabó proved that Γ(t) contains no copy of K t,(t−1)!+1 . Our aim here is to show that it also contains no K t+1,(t−1)!−1 , generalizing the same result for t = 5 presented in [2].…”

The History of Degenerate (Bipartite) Extremal Graph Problems

“…which was slightly improved by Znám [257], [256], (he halved the last term to (a − 1)n/2 in the case of m = n) and Guy [128]. 19 In other terminology, the hypergraph defined by the rows of this matrix is 18 In Graph Theory two problems are connected to Zarankiewicz' name: the extremal problem for matrices that we shall discuss here and the Crossing Number conjecture which is not our topic. Actually, the crossing number problem comes from Paul Turán, see [244].…”

On Sets Free of Sumsets with Summands of Prescribed Size

“…Füredi [15] obtained (1.7) and Brown [5] and Füredi [15] proved (1.8), whereas (1.9) was proved by Alon, Rónyai and Szabó [2]. Ball and Pepe [3] have recently proved that ex(n, K 5,5 ) ≫ n 7/4 . Their result also improves the exponent in the lower bound in (1.5) for the cases (ℓ 1 , ℓ 2 ) = (5, ℓ 2 ), 5 ≤ ℓ 2 ≤ 12 and for (ℓ 1 , ℓ 2 ) = (6, ℓ 2 ), 6 ≤ ℓ 2 ≤ 8.…”