Noticias en español
Chen-Chvátal Conjecture in Graphs and Hypergraphs.
Abstract: It is well known that a set of n non-collinear points in the Euclidean plane determines at least n distinct lines. In 2008, Chen and Chvátal conjectured that this result extends to arbitrary finite metric spaces with an appropriate definition of line. In this talk, we present a survey of this conjecture, outlining known results in the contexts of metric spaces, hypergraphs, and graphs
Read MoreThe Brown-Erdős-Sós conjecture in dense triple systems.
Abstract: In 1973, Brown, Erdős, and Sós conjectured, in an equivalent form, that for any $\delta > 0$ and integer $e \geq 3$, every sufficiently large linear 3-uniform hypergraph with at least $\delta n^2$ edges contains a collection of $e$ edges whose union spans at least $e+3$ vertices. In this talk, we show a proof of the conjecture for the case $\delta > 4/5$.
Read MoreThe Ramsey Number of Multiple Copies of a Graph
Abstract: Let H be a graph without isolated vertices. The Ramsey Number r(nH) is the minimum N such that every coloring of the edges of the complete graph on N vertices with red and blue contains n pairwise vertex-disjoint monochromatic copies of H of the same color. In 1975, Burr, Erdős and Spencer established that r(nH) is a linear function of n for large enough n. In 1987, Burr proved a superexponential upper bound for when the long-term linear behavior starts. In 2023, Bucic and Sudakov showed that the long-term linear behavior starts already when n is an exponential function of |H|...
Read MoreThe Ramsey Number of Hypergraph Cycles.
Abstract: In 1999, Łuczak proved that the three-coloured Ramsey number of the cycle of length $n$ is less than $(4+o(1))n$, presenting a technique that, conceptually, through the use of Szemerédi’s regularity lemma, reduces the problem to that of finding the Ramsey number of a connected matching. Ever since then, Łuczak’s method has been applied successfully in many results. There are many natural ways to generalize cycles for $k$-uniform hypergraphs. In this talk, we first present a brief survey of the Ramsey numbers of Berge, loose, and tight cycles. Then we examine the use...
Read MoreAbstract: For a family of graphs H, a graph G is H-free if no induced subgraph of G is isomorphic to a graph in H. In this talk, I will present a new decomposition theorem and coloring algorithm for(2P_3,C_4,C_6)-free graphs. I will also give some background on Truemper configurations (thetas, pyramids, prisms, and wheels) and on proving decomposition theorems in general
Read MoreCanonical Ramsey numbers for partite hypergraphs.
Abstract: Ramsey’s theorem states that for sufficiently large n, any r-colouring of the edges of the complete k-uniform hypergraph on n vertices contains a monochromatic copy of the complete k-uniform hypergraph on t vertices. Erdős and Rado generalised this result to an unbounded number of colours. They characterised all canonical colour patterns that are unavoidable in edge colourings of sufficiently large complete hypergraphs. We consider quantitative aspects of this result. For Erdős and Rado’s theorem, both the lower and upper bound on n grow as (k-1)-times iterated...
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