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A disproof of a conjecture of Erdős in Ramsey theory
 J. London Math. Soc
, 1989
"... Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of monochrom ..."
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Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of monochromatic Kt's in a random colouring of Kn. We present counterexamples to this conjecture and discuss properties of the extremal graphs. 1.
ONEDIMENSIONAL ASYMPTOTIC CLASSES OF FINITE STRUCTURES
"... Abstract. A collection C of finite Lstructures is a 1dimensional asymptotic class if for every m ∈ N and every formula ϕ(x, ¯y), where ¯y =(y1,...,ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈Cand ā ∈ Mm,eitherϕ(M,ā)  ≤C,orforsomeµ ∈ E, ..."
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Cited by 9 (0 self)
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Abstract. A collection C of finite Lstructures is a 1dimensional asymptotic class if for every m ∈ N and every formula ϕ(x, ¯y), where ¯y =(y1,...,ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈Cand ā ∈ Mm,eitherϕ(M,ā)  ≤C,orforsomeµ ∈ E,
Random Strongly Regular Graphs?
 Discrete Math
"... Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36,10,4,2), but there are 32548 nonisomorphic graphs with parameters (36,15,6,6). (The first assertion is a special case of a theorem of Shr ..."
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Cited by 8 (1 self)
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Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36,10,4,2), but there are 32548 nonisomorphic graphs with parameters (36,15,6,6). (The first assertion is a special case of a theorem of Shrikhande, while the second is the result of a computer search by McKay and Spence.) In the light of this, it will be difficult to develop a theory of random strongly regular graphs! For certain values of the parameters, we have at least one prerequisite for a theory of random objects: there should be very many of them (e.g. superexponentially many). Two other features we would like are a method to sample from the uniform distribution (this is known in a couple of special cases) and information about how various graph parameters behave as random variables on the uniform distribution. Very little is known but there are a few recent results and some interesting problems. This paper dev...
A Prolific Construction of Strongly Regular Graphs With the NE.c. Property
, 2002
"... A graph is ne.c. (nexistentially closed) if for every pair of subsets U , W of the vertex set V of the graph such that U \W = ; and jU j + jW j = n, there is a vertex v 2 V (U [ W ) such that all edges between v and U are present and no edges between v and W are present. A graph is strongly reg ..."
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Cited by 7 (0 self)
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A graph is ne.c. (nexistentially closed) if for every pair of subsets U , W of the vertex set V of the graph such that U \W = ; and jU j + jW j = n, there is a vertex v 2 V (U [ W ) such that all edges between v and U are present and no edges between v and W are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices v 1 ; v 2 2 V depends only on whether or not fv 1 ; v 2 g is an edge in the graph. The only
Combinatorics of Monotone Computations
 Combinatorica
, 1998
"... Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This r ..."
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Cited by 6 (0 self)
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Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion...
Explicit graphs with extension properties
 Bulletin of the European Association for Theoretical Computer Science
"... Abstract. We exhibit explicit, combinatorially defined graphs satisfying the k th extension axiom: Given any set of k distinct vertices and any partition of it into two pieces, there exists another vertex adjacent to all of the vertices in the first piece and to none in the second. Quisani: 1 I’ve b ..."
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Cited by 4 (3 self)
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Abstract. We exhibit explicit, combinatorially defined graphs satisfying the k th extension axiom: Given any set of k distinct vertices and any partition of it into two pieces, there exists another vertex adjacent to all of the vertices in the first piece and to none in the second. Quisani: 1 I’ve been reading about zeroone laws, and many of the results involve extension axioms. In the simple case of graphs, by which I mean undirected graphs without loops or multiple edges, the k th extension axiom 2 says that, for any k distinct vertices x1,..., xk and any subset S ⊆ {1,..., k}, there is another vertex adjacent to xα for all α ∈ S and for no other α. I know that each of these axioms is true in almost all sufficiently large finite graphs. (Of course, “sufficiently large ” depends on k.) So there are lots of these graphs, but I’d like to see some actual examples. Authors: Well, just take a big set of vertices, flip coins to decide
Graphs With The 3e.c. Adjacency Property Constructed From Affine Planes
 J. COMB. MATH. COMB. COMP
"... A graph G is 3e.c. if for each distinct triple S of vertices, and each subset T of S, there is a vertex not in S joined to the vertices of T and to no other vertices of S. Few explicit examples of 3e.c. graphs are known, although almost all graphs are 3e.c. We provide new examples of 3e.c. g ..."
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Cited by 3 (1 self)
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A graph G is 3e.c. if for each distinct triple S of vertices, and each subset T of S, there is a vertex not in S joined to the vertices of T and to no other vertices of S. Few explicit examples of 3e.c. graphs are known, although almost all graphs are 3e.c. We provide new examples of 3e.c. graphs arising as incidence graphs of partial planes resulting from affine planes. We also present a new graph operation that preserves the 3e.c. property.
GRAPHS WITH THE nE.C. ADJACENCY PROPERTY Constructed From Affine Planes
"... We give new examples of graphs with the ne.c. adjacency property. Few explicit families of ne.c. graphs are known, despite the fact that almost all finite graphs are ne.c. Our examples are incidence graphs of certain partial planes derived from a#ne planes of even order. We use probabilistic ..."
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We give new examples of graphs with the ne.c. adjacency property. Few explicit families of ne.c. graphs are known, despite the fact that almost all finite graphs are ne.c. Our examples are incidence graphs of certain partial planes derived from a#ne planes of even order. We use probabilistic and geometric techniques to construct new examples of ne.c. graphs from partial planes for all n, and we use geometric techniques to give new examples with small orders if n = 3. We give a new construction, using switching, of an exponential number of nonisomorphic ne.c. graphs on