Let [n]r be the complete r-partite hypergraph with vertex classes of size n. It is an easy exercise to show that every set of more than (k − 1)nr−1 edges in [n]r contains a matching of size k. We conjecture the following rainbow version of this observation: If F1, F2, . . . , Fk ⊆ [n]r are of size larger than (k− 1)nr−1 then there exists a rainbow matching, i.e. a choice of disjoint edges fi ∈ ...

J. Song, Y. Huang, J. Xiao, S. Wang, K. C. Hwang, H. C. Ko, D.-H. Kim, M. P. Stoykovich, and J. A. Rogers Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, Florida 33146, USA Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 60208, USA Department of Mechanical Engineering, Northwestern University, Evanston, Illino...

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko’s framework for polynomial space computability in R to define weakly pspace-random points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem characterizes weakly pspace random points. That is, a po...

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko’s framework for polynomial space computability in Rn to define weakly pspace-random points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem holds for every weakly pspace-random point.

We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random kuniform hypergraph H(n, p). For 2 ≤ k(n) ≤ n/2, let N = ( n k ) and D = ( n−k k ) . We show that with probability tending to 1 as n→∞, the largest intersecting subhypergraph of H(n, p) has size (1 + o(1))p k nN , for any p n k ln 2 ( n k ) D−1. This lower bound on p is asymptotically best possible for k = Θ(n). F...

We define and compare several probabilistic notions of computability for mappings from represented spaces (that are equipped with a measure or outer measure) into computable metric spaces. We thereby generalize definitions by [Ko 1991] and Parker (see [Parker 2003, Parker 2005, Parker 2006]), and furthermore introduce the new notion of computability in the mean. Some results employ a notion of ...

The ordering principle states that every finite linear order has a least element. We show that, in the relativized setting, the surjective weak pigeonhole principle for polynomial time functions does not prove a Herbrandized version of the ordering principle over T 2 . This answers an open question raised in [Buss, Ko lodziejczyk and Thapen, 2012] and completes their program to compare the stre...

Let n, d, and k be positive integers such that k ≥ 2, d ≥ 1, and n = kd. Furthermore, let N = {1, . . . , n} be the ground set and B(n, d) = {S ⊆ N : |S| = d} denote the family of all subsets of cardinality d in N ; obviously, |B(n, d)| = ( n d ) . A subfamily P (n, d) ⊆ B(n, d) that consists of k pairwise disjoint subsets is called a partition. Obviously, all sets of B(n, d) participate the sa...

A graph G is called (k, d)∗-choosable if, for every list assignment L satisfying |L(v)| = k for all v ∈ V (G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Ko-Wei Lih et al. used the way of discharging to prove that every planar graph without 4-cycles and i-cycles for some i ∈ {5, 6, 7} is (3, 1)∗-choosable. In this paper,...

Some Erdős-Ko-Rado type extremal properties of families of vectors from {−1, 0, 1} are considered.