The rado multiplicity problem in vector spaces over finite fields

We study an analogue of the Ramsey multiplicity problem for additive structures, establishing the minimum number of monochromatic $3$-APs in $3$-colorings of $\mathbb{F}_3^n$ and obtaining the first non-trivial lower bound for the minimum number of monochromatic $4$-APs in $2$-colorings of $\mathbb{F}_5^n$. The former parallels results by Cumings et al.~\cite{CummingsEtAl_2013} in extremal graph theory and the latter improves upon results of Saad and Wolf~\cite{SaadWolf_2017}. Lower bounds are notably obtained by extending the flag algebra calculus of Razborov~\cite{razborov2007flag}.Peer ReviewedPostprint (author's final draft

The Ramsey number of books

We show that in every two-colouring of the edges of the complete graph K_N there is a monochromatic K_k which can be extended in at least (1 + o_(k)(1))2^(-k)N ways to a monochromatic K_(k+1). This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book B_n^(k) to be the graph consisting of n copies of K_(k+1) all sharing a common K_k, we show that the Ramsey number r(B_n^(k)) = 2^(k)n + o_(k)(n). In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason

Monochromatic triangles in three-coloured graphs

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.Comment: Some data needed to verify the proof can be found at http://www.math.cmu.edu/users/jcumming/ckpsty

Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from $\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we prove that for any positive integers $d$ and $k$, there exists a constant $c$ such that if $H$ is an $n$-vertex graph with maximum degree $d$, then $GR_k(H)$ is at most $cn$. We also determine $GR_k(K_4+e)$ for the graph on 5 vertices consisting of a $K_4$ with a pendant edge. Furthermore, we consider two extremal problems related to Gallai-$k$-colorings. For $n\geq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, implying that this number is $O(n^3)$ and yielding the exact value for $k=3$. We also determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$.Comment: 20 pages, 1 figur

String graphs and incomparability graphs

Given a collection C of curves in the plane, its string graph is defined as the graph with vertex set C, in which two curves in C are adjacent if and only if they intersect. Given a partially ordered set (P,<), its incomparability graph is the graph with vertex set P, in which two elements of P are adjacent if and only if they are incomparable. It is known that every incomparability graph is a string graph. For “dense” string graphs, we establish a partial converse of this statement. We prove that for every ε>0 there exists δ>0 with the property that if C is a collection of curves whose string graph has at least ε|C|[superscript 2] edges, then one can select a subcurve γ′ of each γ∈C such that the string graph of the collection {γ′:γ∈C} has at least δ|C|[superscript 2] edges and is an incomparability graph. We also discuss applications of this result to extremal problems for string graphs and edge intersection patterns in topological graphs.National Science Foundation (U.S.). Graduate Research FellowshipPrinceton University (Centennial Fellowship)Simons FoundationNational Science Foundation (U.S.) (Grant DMS-1069197

The Ramsey number of books

We show that in every two-colouring of the edges of the complete graph K_N there is a monochromatic K_k which can be extended in at least (1 + o_(k)(1))2^(-k)N ways to a monochromatic K_(k+1). This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book B_n^(k) to be the graph consisting of n copies of K_(k+1) all sharing a common K_k, we show that the Ramsey number r(B_n^(k)) = 2^(k)n + o_(k)(n). In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason