On linear graph invariants related to Ramsey and edge numbers : or how I learned to stop worrying and love the alien invasion

• In this thesis we study the Ramsey numbers, R(l,k), the edge numbers, e(l,k;n) and graphs that are related to these. The edge number e(l,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 l or an independent set of size k. The Ramsey number R(l,k) may then be defined as the least natural number n for which e(l,k;n) = ∞ .In Paper I, IV and V we study strict lower bounds for e(l,k;n). In Paper I we do this in the case where l = 3 by, in particular, showing e(G) ≥ (1/3)(17n(G) - 35α(G) - N(C4;G)) for all triangle-free graphs G, where N(C4;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) ≥ (1/2)(ceil((7l - 2)/2)n(G) - l floor((5l + 1)/2)α(G))for l ≥ 2, and examine the properties of graphs G without cliques of size l+1 such that G is minimal with respect to the above inequality not holding, and show for small l 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 when n ≤ 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 performed 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(l,k). In particular the method uses bounds for the edge numbers e(l,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(l,k).

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Diskret matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Discrete Mathematics (hsv//eng)

Keyword

Ramsey number

edge number

minimal Ramsey graph

independence number

clique number

Turán's theorem

crochet pattern

H13-pattern

linear graph invariant

triangle-free graph

Mathematics

matematik

Publication and Content Type

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