The existence of unavoidable repeated substructures is a known phenomenon implied by the pigeonhole principle and its generalizations. A fundamental problem is to determine the largest size of a repeated substructure in any combinatorial structure from a given class. The strongest notion of repetition is a pair of isomorphic substructures, such as a pair of vertexdisjoint or edge-disjoint isomo...

We consider the clause{based version of the general model of semantic derivations proposed by Kraj cek. Resolution refutation proof is a special deterministic version of fanin-2 clause{based derivation. We prove the following combinatorial lower bound on the length of such derivations. Let F be a k-partite hypergraph, with at most b points in each part such that no point belongs to more than d ...

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov ’03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We fur...

Edward Early ([email protected]) received his Ph.D. from the Massachusetts Institute of Technology. He is an associate professor of mathematics at St. Edward’s University in Austin, TX. His research is mostly in combinatorics, but he enjoys dabbling in number theory. He also likes spending time with his family and training in martial arts. Patrick Kim ([email protected]) is a senior at S...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed be Borel, or even analytic. One the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff packing dimensions, then $$ \dim_{\mathrm{H}} \pi_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} fo...

The cutting plane refutation system CP for propositional logic is an extension of resolution and is based on showing the non-existence of solutions for families of integer linear inequalities. We deene the system CP + , a modiication of the cutting plane system, and show that CP + can polynomially simulate Frege systems F. In 8], it is shown that F polynomially simulates CP + , so both systems ...

Inspired by Ramsey’s theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph G G </mml:...

A binary structure S has the pigeonhole property (P) if every finite partition of S induces a block isomorphic to S. We classify all countable tournaments with (P); the class of orders with (P) is completely classified.