Li, Nikiforov and Schelp [12] conjectured that any 2-edge coloured graph G with order n and minimum degree Î´(G) > 3n/4 contains a monochromatic cycle of length `, for all ` âˆˆ [4, dn/2e]. We proveâ€¦ (More)

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and ErdÅ‘s conjectured that whenâ€¦ (More)

We consider a classical model known as bootstrap percolation on the nÃ—n square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds weâ€¦ (More)

In this paper we study multipartite Ramsey numbers for odd cycles. Our main result is the proof of a conjecture of GyÃ¡rfÃ¡s, SÃ¡rkÃ¶zy and Schelp [12]. Precisely, let n â‰¥ 5 be an arbitrary positive oddâ€¦ (More)

Bootstrap percolation, one of the simplest cellular automata, can be seen as a model of the spread of infection. In r-neighbour bootstrap percolation on a graph G we assign a state, infected orâ€¦ (More)