More on Rainbow Cliques in Edge-Colored Graphs

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any $\varepsilon>0$, if $e(G)+c(G) \geq (1+\frac{k-3}{k-2}+2\varepsilon) {n\choose 2}$ and $k\geq 3$, then for sufficiently large $n$, the number of rainbow cliques $K_k$ in $G$ is $\Omega(n^k)$. We also characterize the extremal graphs $G$ without a rainbow clique $K_k$, for $k=4,5$, when $e(G)+c(G)$ is maximum. Our results not only address existing questions but also complete the findings of Ehard and Mohr (Ehard and Mohr, Rainbow triangles and cliques in edge-colored graphs. {\it European Journal of Combinatorics, 84:103037,2020}).Comment: 16page

Note on vertex disjoint rainbow triangles in edge-colored graphs

Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow triangles. Rainbow problems have been studied extensively in the context of anti-Ramsey theory, and more recently, in the context of Tur\'{a}n problems. B. Li. et al. \textit{European J. Combin. 36 (2014)} found that a graph must contain a rainbow triangle if $e(G)+c(G) \geq \binom{n}{2}+ n$. L. Li. and X. Li. \textit{Discrete Applied Mathematics 318 (2022)} conjectured a lower bound on $e(G)+c(G)$ such that $G$ must contain a proper $mK_3$. In this paper, we provide a construction that disproves the conjecture. We also introduce a result that guarantees the existence of $m$ vertex disjoint rainbow $K_k$ subgraphs in general host graphs, and a sharp result on the existence of proper $mK_3$ in complete graphs

On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

Using Ramsey theory to measure unavoidable spurious correlations in Big Data

Given a dataset we quantify how many patterns must always exist in the dataset. Formally this is done through the lens of Ramsey theory of graphs, and a quantitative bound known as Goodman's theorem. Combining statistical tools with Ramsey theory of graphs gives a nuanced understanding of how far away a dataset is from random, and what qualifies as a meaningful pattern. This method is applied to a dataset of repeated voters in the 1984 US congress, to quantify how homogeneous a subset of congressional voters is. We also measure how transitive a subset of voters is. Statistical Ramsey theory is also used with global economic trading data to provide evidence that global markets are quite transitive.Comment: 21 page