Ramsey numbers of Berge-hypergraphs and related structures

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.Comment: Updated to include suggestions of the refere

Avoiding long Berge cycles

Let n≥k≥r+3 and H be an n-vertex r-uniform hypergraph. We show that if |H|>[Formula presented](k−1r) then H contains a Berge cycle of length at least k. This bound is tight when k−2 divides n−1. We also show that the bound is attained only for connected r-uniform hypergraphs in which every block is the complete hypergraph Kk−1 (r). © 201

Connections Between Extremal Combinatorics, Probabilistic Methods, Ricci Curvature of Graphs, and Linear Algebra

This thesis studies some problems in extremal and probabilistic combinatorics, Ricci curvature of graphs, spectral hypergraph theory and the interplay between these areas. The first main focus of this thesis is to investigate several Ramsey-type problems on graphs, hypergraphs and sequences using probabilistic, combinatorial, algorithmic and spectral techniques: The size-Ramsey number Rˆ(G, r) is defined as the minimum number of edges in a hypergraph H such that every r-edge-coloring of H contains a monochromatic copy of G in H. We improved a result of Dudek, La Fleur, Mubayi and Rödl [ J. Graph Theory 2017 ] on the size-Ramsey number of tight paths and extended it to more colors. An edge-colored graph G is called rainbow if every edge of G receives a different color. The anti-Ramsey number of t edge-disjoint rainbow spanning trees, denoted by r(n, t), is defined as the maximum number of colors in an edge-coloring of Kn containing no t edge-disjoint rainbow spanning trees. Confirming a conjecture of Jahanbekam and West [J. Graph Theory 2016], we determine the anti-Ramsey number of t edge-disjoint rainbow spanning trees for all values of n and t. We study the extremal problems on Berge hypergraphs. Given a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection i ∶ V (G) → V (H) and a bijection f ∶ E(G) → E(H) such that for every e = uv ∈ E(G), (i(u), i(v)) ⊆ f(e). We investigate the hypergraph Ramsey number of Berge cliques, the cover-Ramsey number of Berge hypergraphs, the cover-Turán desity of Berge hypergraphs as well as Hamiltonian Berge cycles in 3-uniform hypergraphs. The second part of the thesis uses the ‘geometry’ of graphs to derive concentration inequalities in probabilities spaces. We prove an Azuma-Hoeffding-type inequality in several classical models of random configurations, including the Erdős-Rényi random graph models G(n, p) and G(n,M), the random d-out(in)-regular directed graphs, and the space of random permutations. The main idea is using Ollivier’s work on the Ricci curvature of Markov chairs on metric spaces. We give a cleaner form of such concentration inequality in graphs. Namely, we show that for any Lipschitz function f on any graph (equipped with an ergodic random walk and thus an invariant distribution ν) with Ricci curvature at least κ \u3e 0, we have ν (∣f − Eνf∣ ≥ t) ≤ 2 exp (-t 2κ/7). The third part of this thesis studies a problem in spectral hypergraph theory, which is the interplay between graph theory and linear algebra. In particular, we study the maximum spectral radius of outerplanar 3-uniform hypergraphs. Given a hypergraph H, the shadow of H is a graph G with V (G) = V (H) and E(G) = {uv ∶ uv ∈ h for some h ∈ E(H)}. A 3-uniform hypergraph H is called outerplanar if its shadow is outerplanar and all faces except the outer face are triangles, and the edge set of H is the set of triangle faces of its shadow. We show that the outerplanar 3-uniform hypergraph on n vertices of maximum spectral radius is the unique hypergraph with shadow K1 + Pn−1