Ramsey-goodness and otherwise

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number ∆, there is a constant r ∆ such that, for any connected n-vertex graph G with maximum degree ∆, the Ramsey number R(G, G) is at most r ∆ n, provided n is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take r ∆ = ∆. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r ∆ > 2 c∆ for some constant c > 0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n) = o(n), then R(G, G) ≤ (2χ(G) + 4)n ≤ (2∆ + 6)n, i.e., r ∆ = 2∆ + 6 suffices. On the other hand, we show that Burr's conjecture itself fails even for P k n , the kth power of a path P n . Brandt showed that for any c, if ∆ is sufficiently large, there are connected nvertex graphs G with ∆(G) ≤ ∆ but R(G, K 3 ) > cn. We show that, given ∆ and H, there are β > 0 and n 0 such that, if G is a connected graph on n ≥ n 0 vertices with maximum degree at most ∆ and bandwidth at most βn, then we have R(G, H) = (χ(H) − 1)(n − 1) + σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ε(H) log n/ log log n

Ramsey numbers of squares of paths

The Ramsey number R(G;H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G;H) of graphs. The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths. The r-th power Pr n of a path on n vertices is obtained by joining any two vertices with distance at most r. We determine the exact value of R(P2 n; P2 n) for n large and discuss some related questions

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding results for graphs embeddable on different orientable surfaces.Comment: 20 page

Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)- coloring of H. A graph G is called H-good if R(G, H) = (|G| − 1)(χ(H) − 1) + σ(H). The notion of Ramsey goodness was introduced by Burr and Erd˝os in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n ≥ 4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan