Norm-Graphs: Variations and Applications

Abstract

We describe several variants of the norm-graphs introduced by Kolla r, Ro nyai, and Szabo and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 1 2 n 5Â3 edges, containing no copy of K 3, 3 , thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K 3, 3 is (1+o(1)) k 3 . This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose so-called dual shatter function is O(m t ) and whose discrepancy is 0(n 1Â2&1Â2t -log n). This settles a problem of Matous ek.

Counting configuration–free sets in groups

“…We compute both the values α k i (for i ∈ [1,8]) and min |U|=i,U⊂ [1,k] On the other hand, there are projections π U that have more values than what they should. This forces the probability in Proposition 5.5 to be smaller and compensate for this.…”

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

“…Conjecture 1 is then equivalent to asking how large an ordered bipartite graph can be while avoiding an ordered matching. Let H(P ) be the unordered (bipartite) graph corresponding to matrix P and let Ex T u (H, n) be the Turán-number of a graph H, i.e., the maximum number of edges in an n-vertex graph avoiding subgraphs isomorphic to H. At a high level the growth of Ex T u (H, n) is understood very well; it is Θ(n 2 ) if H is not bipartite, O(n) if H is a forest, and Ω(n 1+c1 ) and O(n 1+c2 ) in all other cases, for constants 0 < c 1 ≤ c 2 < 1 [7,13,22,2,6,5]. Füredi and Hajnal conjectured [14] that the extremal functions for 0-1 matrix avoidance and unordered subgraph avoidance never differ by more than a logarithmic factor:…”

“…(Determining Ex(K k,l , n) is sometimes called Zarankiewicz's problem [7,13,22,2].) Assume without loss of generality that l ≥ k. Kövari, Sós, and Turán [22] proved that Ex T u (K k,l ) = O(n 2−1/k ) and it is widely believed that this is the correct bound for fixed k and l. However, the upper bound has only been proved tight when k ∈ {1, 2, 3} (with ever sharper bounds on the leading constants and lower order terms [9,7,13,12,2]) or if k ≥ 4 and l ≥ (k − 1)! + 1 [2,21].…”

“…Part (3) follows the same lines as part (2). If columns 1 and 4 of X were in the same block then that block (D k ) would include Q 1 , a contradiction; if they are in different blocks and j 2 = j 2 then, depending on which of j 2 and j 2 is larger, D k would include either:…”

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

“…The graphs G q,t were first constructed by Alon, Rónyai and Szabó to give improved lower bounds for the Turán function ex(n, K t,s ) and the Ramsey numbers R k (K t,s ) where t ≥ 2 and s ≥ (t − 1)! + 1 (see [3]). In [22] T. Szabó found the eigenvalues of G q,t to be ±q (t−1)/2 , ±1, 0 and q t−1 − 1.…”