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Oliver Krüger: On linear graph invariants related to Ramsey and edge numbers

Time: Mon 2019-12-16 10.00

Location: Kräftriket, house 5, room 14

Subject area: Combinatorics

Doctoral student: Oliver Krüger , Stockholms universitet

Opponent: Alexander Engström, Aalto University

Supervisor: Jörgen Backelin, Stockholms universitet

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In this thesis we study the Ramsey numbers, \(R(\ell,k)\), the edge numbers, \(e(\ell,k;n)\) and graphs that are related to these. The edge number \(e(\ell,k;n)\) may be defined as the least natural number \(m\) for which all graphs on \(n\) vertices and less than \(m\) edges either contains a complete subgraph of size \(\ell\) or an independent set of size \(k\). The Ramsey number \(R(\ell,k)\) may then be defined as the least natural number \(n\) for which \(e(\ell,k;n) = \infty\).

In Paper I, IV and V we study strict lower bounds for \(e(\ell,k;n)\). In Paper I we do this in the case where \(\ell = 3\) by, in particular, showing \(e(G) \geq \frac{1}{3}\left( 17n(G) - 35\alpha(G) - \mathrm{N}(C_4;G) \right)\) for all triangle-free graphs \(G\), where \(\mathrm{N}(C_4;G)\) denotes the number of cycles of length \(4\) in \(G\). In Paper IV we describe a general method for generating similar inequalities to the one above but for graphs that may contain triangles, but no complete subgraphs of size \(4\). We then show a selection of the inequalities we get from the computerised generation. In Paper V we study the inequality

\(e(G) \geq \frac{1}{2}\left( \left\lceil \frac{7\ell - 2}{2} \right\rceil n(G) - \ell \left\lfloor \frac{5\ell + 1}{2} \right\rfloor \alpha(G) \right)\)

for \(\ell \geq 2\), and examine the properties of \(K_{\ell + 1}\)-free graphs \(G\) such that \(G\) is minimal with respect to the above inequality not holding, and show for small \(\ell\) that no such graphs \(G\) can exist.

In Paper II we study constructions of graphs \(G\) such that \(e(G) - e(3,k;n)\) is small and \(n \leq 3.5(k-1)\). We employ a description of some of these graphs in terms of 'patterns' and a recursive procedure to construct them from the patterns. We also present the result of computer calculations where we actually have preformed such constructions of Ramsey graphs and compare these lists to previously computed lists of Ramsey graphs.

In Paper III we develop a method for computing, recursively, upper bounds for Ramsey numbers \(R(\ell,k)\). In particular the method uses bounds for the edge numbers \(e(\ell,k;n)\). In Paper III we have implemented this method as a computer program which we have used to improve several of the best known upper bounds for small Ramsey numbers \(R(\ell,k)\).