Abstract. We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements. We allow graphons on general probability spaces whenever possible. We also give some new results for {0,1}-valued graphons. 1.

Abstract. The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (Gn) of graphs in terms of a limiting object which may be represented by a symmetric function W on [0, 1], i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a ‘quasi-monotonicity ’ property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and L 1 norms of kernels of the form W1 − W2 with W1 and W2 monotone that may be of interest in its own right; no such inequality holds for general kernels.

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for nite sums and a normal limit theorem as the size tends to in nity. The tails for nite sums may be much larger that the tails of the limit.

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