In the Ramsey theory ofgraphs F + (G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H. The problem ARROWING of deciding whether F + (G, H) lies in II; = coNPNP and it was shown to be coNP hard by Burr [5]. We prove that ARROWING is actually II;-complete, simultaneously settling a conjecture of Burr and providing a natural example of a problem complete for a higher level of the polynomial hierarchy. We also show that STRONG ARROWING, the version for induced subgraphs, is rI;-complete. 1

A conjecture of Loebl, known as the (n/2 − n/2 − n/2) Conjecture, states that if G is an n-vertex graph in which at least n/2 of the vertices have degree at least n/2, then G contains all trees with at most n/2 edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi proved an approximate version of this conjecture. We prove it exactly for sufficiently large n and show that this is essentially best possible. Our result gives a tight bound for the Ramsey number of trees, and thus partially answers a conjecture of Burr and Erdős. Along with our proof, we also give a stability theorem, which describe the structure of n-vertex graphs that have at least (1 − ε)n/2 vertices of degree at least n/2 but does not contain certain tree with n/2 edges. 1.

We prove a version of the Loebl-Komlós–Sós Conjecture for dense graphs.For any q> 0 there exists a number n0 2 N such that for any n> n0 and k> qn the following holds: if G be a graph of order n with at least n

For two given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C,W,,) for m = 4 and m = 5. We show that R(C,W4) = 2n - 1 and R(C, Wx) = 3n - 2 for n _> 5. For larger wheels it remains an open problem to determine R(C, W,,).

It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17; where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels. 1 Introduction The following well known conjecture is due to Paul Erdos. CONJECTURE 1 If G is a graph with chromatic number Ø(G) k; then the Ramsey number r(G) r(K k ): The strong form of the Erdos conjecture is that if Ø(G) k; and G does not contain a copy of K k ; then r(G) ? r(K k ). For k = 3 it is trivial to verify this stronger conjecture. If G 6' K 3 and Ø(G) 3; then G has at least 4 vertices. Thus r(G) ? 6 = r(K 3 ); since neither the graph K 3 [K...

For two given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we shall study the Ramsey number R(T, W,) for a star-like tree T with n vertices and a wheel W, with m + 1 vertices and m odd. We show that the Ramsey number R(S,W,) -- 3n - 2 for n _ 2m - 4, m _ 5 and m odd, where S denotes the star on n vertices. We conjecture that the Ramsey number is the same for general trees on n vertices, and support this conjecture by proving it for a number of star-like trees.

We show that in every r-coloring of the edges of Kn there is a monochromatic double star with at least n(r+1)+r−1 r 2 vertices. This result is sharp in asymptotic for r = 2and for r ≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When r-colorings are replaced by local r-colorings, our bound is n(r+1)+r−1 r 2. +1

integer m such that, for any partition (E1, E2) of the edges of K,,, , either G 1 is a subgraph of the graph induced by E1, or G2 is a subgraph of the graph induced by E2. We show that R(C „ , Cn) = 2n- 1 if n is odd, R(Cs, C2,_,) = 2n- I if n> r(2r- 1), R(C„,C2,)=n+r-1 if n>4r 2-r+2, R(Cn, K,) v nr 2 for all r, n, R(Cn,K,.)=(r-1)(n-1) +1 if n>r2-2, R(Cn, K,+1) = t(n- 1) + r for large n. 1

Chvátal has shown that if T is a tree on n points then r(Kk, T) _ (k- 1) (n- 1) + 1, where r is the (generalized) Ramsey number. It is shown that the same result holds when T is replaced by many other graphs. Such a T is called k-good. The results proved all support the conjecture that any large graph that is sufficiently sparse, in the appropriate sense, is k-good. 1

to be the least number p such that if the edges of the complete graph colored red and blue (say), either the red graph contains Kp are G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. following result is proved: Let G and H have k and I points respectively and have point independence numbers of i and j respectively. Then N- 1 5 r(mG, nH) < N + C, where N = km + In- min(mi, ml) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mG, nH) for various choices of G and H, especially when m = n or G = H. In particular, r(mK3, nK 3) = 3m + 2n when m _> n, m 3 2. 1. Introduction. Let G and H be graphs without isolated points. Following Chvátal and Harary [11, define the Ramsey number r(G, H) to be the least integer n such that if the edges of Kn (the complete graph on n points) are two-colored,