Ongoing fluctuations of neuronal activity have long been considered intrinsic noise that introduces unavoidable and unwanted variability into neuronal processing, which the brain eliminates by averaging across population activity (Georgopoulos et al., 1986; Lee et al., 1988; Shadlen and Newsome, 1994; Maynard et al., 1999). It is now understood, that the seemingly random fluctuations of cortical activity form highly structured patterns, including oscillations at various frequencies, that modulate evoked neuronal responses (Arieli et al., 1996; Poulet and Petersen, 2008; He, 2013) and affect sensory perception (Linkenkaer-Hansen et al., 2004; Boly et al., 2007; Sadaghiani et al., 2009; Vinnik et al., 2012; Palva et al., 2013). Ongoing cortical activity is driven by proprioceptive and interoceptive inputs. In addition, it is partially intrinsically generated in which case it may be related to mental processes (Fox and Raichle, 2007; Deco et al., 2011). Here we argue that respiration, via multiple sensory pathways, contributes a rhythmic component to the ongoing cortical activity. We suggest that this rhythmic activity modulates the temporal organization of cortical neurodynamics, thereby linking higher cortical functions to the process of breathing.

We prove -for sufficiently large n -the following conjecture of Faudree and Schelp:2n − 1 for odd n, 2n − 2 for even n, for the three-color Ramsey numbers of paths on n vertices.

Improving a result of Erdős, Gyárfás and Pyber for large n we show that for every integer r 2 there exists a constant n 0 = n 0 (r) such that if n n 0 and the edges of the complete graph K n are colored with r colors then the vertex set of K n can be partitioned into at most 100r log r vertex disjoint monochromatic cycles.

Let T(r, n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way that for n sufficiently largeholds.

Let T(r, n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way that for n sufficiently large log2 T(r, n) log 2 r-~<8 .-n r 2 ' holds.

A hypergraph is called an r × r grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e., a family of sets {A 1 , . . . , A r , B 1 , . . . , B r } such thatA hypergraph is linear, if |E ∩ F | ≤ 1 holds for every pair of edges.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles.For r ≥ 4 our constructions are almost optimal. These investigations are also motivated by coding theory: we get new bounds for optimal superimposed codes and designs. *

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In this paper a random graph model G Z 2 N ,p d is introduced, which is a combination of fixed torus grid edges in (Z/N Z) 2 and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u, v ∈ (Z/N Z) 2 with graph distance d on the torus grid is p d = c/N d, where c is some constant. We show that, whp, the diameter D(G Z 2 N ,p d ) = Θ(log N ). Moreover, we consider a modified non-monotonous bootstrap percolation on G Z 2 N ,p d . We prove the presence of phase transitions in mean-field approximation and provide fairly sharp bounds on the error of the critical parameters.