Craig Timmons

A counterexample to sparse removal

The Turan number of a graph H , denoted ex ( n , H ) , is the maximum number of edges in an n -vertex graph with no subgraph isomorphic to H . Solymosi (2011) conjectured that if H is any graph and ex ( n , H ) = O ( n α ) where α 1 , then any n -vertex graph with the property that each edge lies in exactly one copy of H has o ( n α ) edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when H is a non-bipartite graph, in particular when H is a triangle, due to Ruzsa and Szemeredi (1978). Using Sidon sets we exhibit infinitely many bipartite graphs H for which the conjecture is false.

Bounds for generalized Sidon sets

Let ? be an abelian group and g ? h ? 2 be integers. A set A ? ? is a C h g ] -set if given any set X ? ? with | X | = h , and any set { k 1 , ? , k g } ? ? , at least one of the translates X + k i is not contained in A . For any g ? h ? 2 , we prove that if A ? { 1 , 2 , ? , n } is a C h g ] -set in Z , then | A | ? ( g - 1 ) 1 / h n 1 - 1 / h + O ( n 1 / 2 - 1 / 2 h ) . We show that for any integer n ? 1 , there is a C 3 3 ] -set A ? { 1 , 2 , ? , n } with | A | ? ( 4 - 2 / 3 + o ( 1 ) ) n 2 / 3 . We also show that for any odd prime p , there is a C 3 3 ] -set A ? F p 3 with | A | ? p 2 - p , which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer h ? 3 , there is a C h h ! + 1 ] -set A ? { 1 , 2 , ? , n } with | A | ? ( c h + o ( 1 ) ) n 1 - 1 / h . A set A is a weak C h g ] -set if we add the condition that the translates X + k 1 , ? , X + k g are all pairwise disjoint. We use the probabilistic method to construct weak C h g ] -sets in { 1 , 2 , ? , n } for any g ? h ? 2 . Lastly we obtain upper bounds on infinite C h g ] -sequences. We prove that for any infinite C h g ] -sequence A ? N , we have A ( n ) = O ( n 1 - 1 / h ( log n ) - 1 / h ) for infinitely many n , where A ( n ) = | A ? { 1 , 2 , ? , n } | .

Covering complete r-graphs with spanning complete r-partite r-graphs

An r-cut of the complete r-uniform hypergraph Krn is obtained by partitioning its vertex set into r parts and taking all edges that meet every part in exactly one vertex. In other words it is the edge set of a spanning complete r-partite subhypergraph of Krn. An r-cut cover is a collection of r-cuts such that each edge of Krn is in at least one of the cuts. While in the graph case r = 2 any 2-cut cover on average covers each edge at least 2-o(1) times, when r is odd we exhibit an r-cut cover in which each edge is covered exactly once. When r is even no such decomposition can exist, but we can bound the average number of times an edge is cut in an r-cut cover between

A Szemerédi–Trotter type theorem, sum-product estimates in finite quasifields, and related results

1+\frac1{r+1}

Orthogonal Polarity Graphs and Sidon Sets

and

Planar polynomials and an extremal problem of Fischer and Matoušek

1+\frac{1+o(1)}{\log r}

Independent Sets in Polarity Graphs

. The upper bound construction can be reformulated in terms of a natural polyhedral problem or as a probability problem, and we solve the latter asymptotically.

Infinite Turán Problems for Bipartite Graphs

Abstract We prove a Szemeredi–Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and Sarkozy on the solvability of the equations a + b = c d and a b + 1 = c d over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields.

Star coloring bipartite planar graphs

Determining the maximum number of edges in an n-vertex C4-free graph is a well-studied problem that dates back to a paper of Erdős from 1938. One of the most important families of C4-free graphs are the Erdős-Renyi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős-Renyi orthogonal polarity graph. Using this isomorphism, we prove that the Petersen graph is a subgraph of every sufficiently large Erdős-Renyi orthogonal polarity graph.

Star coloring planar graphs from small lists

Abstract Let G be a 3-partite graph with k vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matousek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with ( 1 − o ( 1 ) ) k 3 / 2 triangles, and a double counting argument shows that one cannot have more than ( 1 + o ( 1 ) ) k 7 / 4 triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to ( 1 − o ( 1 ) ) k 5 / 3 .