A note on regular Ramsey graphs

A major problem in extremal combinatorics asks to determine the maximal n for which there exists a graph G on n vertices such that G contains no triangles and no independent set of size t. This Ramsey-type problem was settled asymptotically by Kim [6] in 1995, after a long line of research; Kim showed that n = Θ(t 2 / log t). Recently, Bohman [1] gave an alternative proof of Kim's result by analyzing the so-called triangle-free process, as proposed by Erdős, Suen and Winkler [3], which is a natural way of generating a triangle-free graph. Consider now the above problem with the additional constraint that G must be regular. In this short note we show that the same asymptotic results hold up to constant factors. The main ingredient of the proof is a gadget-like construction that transforms a triangle-free graph with no independent set of size t, which is not too far from being regular, into a triangle-free regular graph with no independent set of size 2t.

Our main result can be stated as follows.

Theorem 1.1. There is a positive constant C so that for every natural n there exists a regular triangle-free graph G on n vertices whose independence number satisfies α(G) ≤ C √ n log n.

Denote by R(k, ℓ) the maximal n for which there exists a graph on n vertices which contains neither a complete subgraph on k vertices nor an independent set on ℓ vertices. Let R reg (k, ℓ) denote the maximal n for which there exists a regular graph on n vertices which contains neither a complete subgraph on k vertices nor an independent set on ℓ vertices. Clearly, for every k and ℓ one has

2 Proof of Theorem 1.1

Note first that the statement of the theorem is trivial for small values of n. Indeed, for every n 0 one can choose the constant C in the theorem so that for n ≤ n 0 , C √ n log n ≥ n, implying that for such values of n a graph with no edges satisfies the assertion of the theorem. We thus may and will assume, whenever this is needed during the proof, that n is sufficiently large.

The following well known theorem due to Gale and to Ryser gives a necessary and sufficient condition for two lists of non-negative integers to be the degree sequences of the classes of vertices of a simple bipartite graph. The proof follows easily from the max-flow-min cut condition on the appropriate network flow graph (see e.g. [

then there exists a simple bipartite graph with degree sequence d on each side. In particular, this holds for

Proof. By Theorem 2.1 it suffices to check that for every s, 1 ≤ s ≤ m, s i=1 d i ≤ m i=1 min{d i , s}. Suppose this is not the case and there is some s as above so that

If

Observe that by doing so the left hand side of (2) increases by d 1 -d i , whereas the right hand side increases by at most this quantity, hence (2) still holds with this new value of d i . We can thus assume that

, as the left hand side does not change, whereas the right hand side can only decrease. Moreover, the new sequence still satisfies (1). Thus we may assume that in (2)

gives

Therefore [(a + 1)s -m]d > s 2 , implying that (a + 1)s -m > 0, that is, s > m a+1 , and

The function g(s) = s 2 (a+1)s-m attains its minimum in the range m a+1 < s ≤ m at s = 2m a+1 and its value at this point is 4m (a+1) 2 . We thus conclude from (3) that d > 4m (a+1) 2 and hence that d 1 = ad > 4am (a+1) 2 contradicting the assumption (1). This completes the proof. Proof. Construct a new graph G ′ as follows. Take two copies of G, and color each of these copies by the same equitable coloring using ∆(G) + 1 colors with all color classes of cardinality either ⌊n/(∆(G) + 1)⌋ or ⌈n/(∆(G) + 1)⌉ using the Hajnal-Szemerédi Theorem [4] (see also a shorter proof due to Kierstead and Kostochka [5]). Let 9 . We can thus connect the vertices of C and C ′ using this bipartite graph such that all vertices in C ∪ C ′ have degree d + ∆(G). By following this method for every color class, we create the graph G ′ which is (d + ∆(G))-regular, triangle-free and has no independent set of cardinality 2t -1.

Consider the following randomized greedy algorithm to generate a graph on n labeled vertices with no Hsubgraph for some fixed graph H. Given a set of n vertices, a sequence of graphs {G as the empty graph, and for each 0 < i ≤ t, the graph G (H) i is defined by G (H) i-1 ∪ {e i } where e i is chosen uniformly at random from all unselected pairs of vertices that do not create a copy of H when added to G (H) i-1 . The process terminates at step t, the first time that no potential unselected pair e t+1 exists. This algorithm is called the H-free process.

The K 3 -free process was proposed by Erdős, Suen and Winkler [3] and was further analyzed by Spencer [7]. Recently, Bohman [1] extending and improving previous results, was able to analyze the K 3 -free process and to show that with high probability it passes through an almost regular Ramsey-type graph.

Theorem 2.5 (Bohman [1]). With high probability1 there exists an integer 1 ≤ m = m(n) such that the following proper

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