David S. Gunderson

Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs

Three classes of finite structures are related by extremal properties: complete d-partite d-uniform hypergraphs, d-dimensional affine cubes of integers, and families of 2d sets forming a d-dimensional Boolean algebra. We review extremal results for each of these classes and derive new ones for Boolean algebras and hypergraphs, several obtained by employing relationships between the three classes. Related partition or coloring problems are also studied for Boolean algebras. Density results are given for Boolean algebras of sets all of whose atoms are the same size.

Extremal graphs for intersecting triangles

Abstract It is known that for a graph on n vertices [ n 2 /4] + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex.

Intersection Statements for Systems of Sets

A family ofrsets is called a?-system if any two sets have the same intersection. Denote byF(n, r) the most number of subsets of ann-element set which do not contain a?-system consisting ofrsets. Constructive new lower bounds forF(n, r) are given which improve known probabilistic results, and a new upper bound is given by employing an argument due to Erdo?s and Szemeredi. Another construction is given which shows that for certainn,F(n, 3)?1.551n?2. We also show a relationship between the upper bound forF(n, 3) and the Erdo?s?Rado conjecture on the largest uniform family of sets not containing a?-system.

Extremal Problems for Affine Cubes of Integers

A collection H of integers is called an affine d-cube if there exist d+1 positive integers x0,x1,…, xd so that***** Insert equation here *****We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1,2,…,n} not containing an affine d-cube. In 1892 Hilbert published the first Ramsey-type result for affine d-cubes by showing that, for any positive integers r and d, there exists a least number n=h(d,r) so that, for any r-colouring of {1,2,…,n}, there is a monochromatic affine d-cube. Improvements for upper and lower bounds on h(d,r) are given for d>2.

Independent Arithmetic Progressions in Clique-Free Graphs on the Natural Numbers

We show that if G is a Kr-free graph on N, there is an independent set in G which contains an arbitrarily long arithmetic progression together with its difference. This is a common generalization of theorems of Schur, van der Waerden, and Ramsey. We also discuss various related questions regarding (m, p, c)-sets and parameter words.

Independent Deuber sets in graphs on the natural numbers

We show that for any k, m, p, c, if G is a Kk-free graph on N then there is an independent set of vertices in G that contains an (m, p, c)-set. Hence if G is a Kk-free graph on N, then one can solve any partition regular system of equations in an independent set. This is a common generalization of partition regularity theorems of Rado (who characterized systems of linear equations Ax = 0 a solution of which can be found monochromatic under any finite coloring of N) and Deuber (who provided another characterization in terms of (m, p, c)-sets and a partition theorem for them), and of Ramseys theorem itself.

Triangle-free graphs with the maximum number of cycles

It is shown that for n ? 141 , among all triangle-free graphs on n vertices, the balanced complete bipartite graph K ? n / 2 ? , ? n / 2 ? is the unique triangle-free graph with the maximum number of cycles. Using modified Bessel functions, tight estimates are given for the number of cycles in K ? n / 2 ? , ? n / 2 ? . Also, an upper bound for the number of Hamiltonian cycles in a triangle-free graph is given.

A Ramsey-type problem and the Turán numbers*: A RAMSEY-TYPE PROBLEM AND THE TURÁN NUMBERS

For each n and k, we examine bounds on the largest number m so that for any k-coloring of the edges of Kn there exists a copy of Km whose edges receive at most k 1 colors. We show that for k ffiffiffi p n þ (n), the largest value of m is asymptotically equal to the Turán number t(n,b 2 =kcÞ, while for any constant > 0, if k ð1 Þ ffiffiffi p n then the largest m is asymptotically larger than that Turán number. 2002 Wiley Periodicals, Inc. J Graph Theory 40: 120–129, 2002

Extremal Numbers for Odd Cycles

We describe the C 2k+1-free graphs on n vertices with maximum number of edges. The extremal graphs are unique for n ∉ {3k − 1, 3k, 4k − 2, 4k − 1}. The value of ex(n, C 2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobas [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n 0(C 2k+1) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph K ? n / 2 ? , ? n / 2 ? contains more cycles than any other n -vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k . For k = 1 , we show that any such counterexamples have n ? 91 and are not homomorphic to C 5 ; and for any fixed k there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a # P -complete problem in general) in a special case used by our bounds.