Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

Let KN denote the complete graph on N vertices with vertex set V = V (KN ) and edge set E = E(KN ). For x, y ∈ V , let xy denote the edge between the two vertices x and y. Let L be any finite set and M ⊆ L3. Let c : E → L. Let [n] denote the integer set {1, 2, . . . , n}. For x, y, z ∈ V , let c(xyz) denote the ordered triple (c(xy), c(yz), c(xz)). We say that c is good with respect to M if the following conditions obtain: (i) ∀x, y ∈ V and ∀(c(xy), j, k) ∈ M, ∃z ∈ V such that c(xyz) = (c(xy), j, k); (ii) ∀x, y, z ∈ V , c(xyz) ∈ M; and (iii) ∀x ∈ V ∀` ∈ L ∃ y ∈ V such that c(xy) = `. We investigate particular subsets M ⊆ L3 and those edge colorings of KN which are good with respect to these subsets M. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture. 1 Motivation and background Let KN denote the complete graph on N vertices with vertex set V = V (KN) and edge set E = E(KN). For x, y ∈ V , let xy denote the edge between the two vertices x and the electronic journal of combinatorics 15 (2008), #R49 1 y. Let L be any finite set and M ⊆ L. Let c : E → L. Let [n] denote the integer set {1, 2, . . . , n}. For x, y, z ∈ V , let c(xyz) denote the ordered triple (c(xy), c(yz), c(xz)). We say that c is good with respect to M if the following conditions obtain: (i) ∀x, y ∈ V and ∀(c(xy), j, k) ∈ M, ∃z ∈ V such that c(xyz) = (c(xy), j, k); (ii) ∀x, y, z ∈ V , c(xyz) ∈ M; and (iii) ∀x ∈ V ∀` ∈ L ∃ y ∈ V such that c(xy) = `. If K = KN has a coloring c which is good with respect to M, then we say that K realizes M (or that M is realizable). If we take Rα = {(x, y) : c(xy) = α}, and let | stand for ordinary composition of binary relations, ie. Rα|Rβ := {(x, z) : ∃y (x, y) ∈ Rα, (y, z) ∈ Rβ}, then conditions (i) and (ii) imply (Rα|Rβ) ∩ Rγ 6= ∅ =⇒ Rγ ⊆ Rα|Rβ. Conditions (i) (iii) are given in [1] where the author calls a coloring on KN that realizes some M a symmetric color scheme. It is proved in [2] that if M is a set of triples that is closed under permutation such that there is at least one α ∈ L such that for all β, γ ∈ L, (α, β, γ) ∈ M, then M is realized by a coloring on Kω, the complete graph on countably many vertices. Any such color α is called a flexible color, since it can participate in any triple. Conditions (i) (iii) may seem quite stringent, but in fact these conditions are satisfied in many natural situations. Recall the notation for the Ramsey numbers; that is, R(k1, k2, . . . , k`) is the minimum integer n such that in any `-coloring of the edges of Kn there is a monochromatic complete graph on kj vertices in color j for some j. In particular, the coloring of K5 which shows R(3, 3) ≥ 6 satisfies (i) (iii), as does the coloring of K8 which shows R(4, 3) ≥ 9, the colorings of K16 that show R(3, 3, 3) ≥ 17, both “twisted” and “untwisted”, and the coloring of K29 given in [7] and [4] that shows that R(4, 3, 3) ≥ 30. In fact, the coloring of K5 without monochromatic triangles is a realization of M0 = {(r, b, b), (b, r, b), (b, b, r), (r, r, b), (r, b, r), (b, r, r)}. The coloring of K8 mentioned above is a realization of M = M0 ∪ {(r, r, r)}; the col orings of K16 are realizations of M = {r, b, g}\{(r, r, r), (b, b, b), (g, g, g)}; the coloring of K29 is a realization of M = {r, b, g}\{(b, b, b), (g, g, g)}. In [1], Comer introduces the number r(k) which is the largest N such that there is a coloring on KN that realizes M = { r1, ..., rk} \{(ri, ri, ri) : i ∈ [k]}. Clearly, r(k) ≤ R( k times { }} { 3, 3, . . . , 3)−1; equality holds for k = 2 and k = 3. An interesting open problem is whether equality holds for all values of k. the electronic journal of combinatorics 15 (2008), #R49 2 Realizations of color schemes arise in connection with projective planes as well. Let L = {r1, . . . , r`}, and let M` = {(ri, rj, rk) : |{i, j, k}| ∈ {1, 3}} . Lyndon proved in [5] that M` is realizable in some complete graph if and only if there exists a projective plane of order `−1, for ` > 2. This result has been extremely important in the theory of relation algebras. In [3], Maddux, Jipsen and Tuza show that for M = L, KN realizes M for arbitrarily large finite N . In the case when M = L, every color in L is a flexible color.