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Given integers n ≥ k > l ≥ 1 and a k-graph F with |V(F)| divisible by n, define t k l (n, F) to be the smallest integer d such that every k-graph H of order n with minimum l-degree δl(H) ≥ d contains an F-factor. A classical theorem of Hajnal and Szemerédi in (Proof of a Conjecture of P. Erd˝os, pp. 601–623, 1969) implies that t2 1 (n, Kt) = (1 − 1/t)n for integers t. For k ≥ 3, t k k−1(n, Kk k ) (the δk−1(H) threshold for perfect matchings) has been determined by Kühn and Osthus in (J Graph Theory 51(4):269–280, 2006) (asymptotically) and Rödl et al. in (J Combin Theory Ser A 116(3):613–636, 2009) (exactly) for large n. In this paper, we generalise the absorption technique of Rödl et al. in (J Combin Theory Ser A 116(3):613–636, 2009) to F-factors. We determine the asymptotic values of t k 1 (n, Kk k (m)) for k = 3, 4 and m ≥ 1. In addition, we show that for t > k = 3 and γ > 0, t3 2 (n, K3 t ) ≤ (1− 2 t2−3t+4 +γ )n provided n is large and t|n. We also bound t 3 2 (n, K3 t )from below. In particular, we deduce that t 3 2 (n, K3 4 ) = (3/4+o(1))n answering a question of Pikhurko in (Graphs Combin 24(4):391–404, 2008). In addition, we prove that t k k−1(n, Kk t ) ≤ (1 − t−1 k−1 −1 + γ )n for γ > 0, k ≥ 6 and t ≥ (3 + √ 5)k/2 provided n is large and t|n.

In this study we investigate the inversion of long period regional waveforms for moment tensors with particular emphasis on aspects important for monitoring. We begin with moment tensor inversions for the September 14, 1988 US-Soviet Joint Verification Experiment (JVE) nuclear test at the Semipalatinsk test site in Eastern Kazakhstan, and several nuclear explosions conducted less than ten years later at the Chinese Lop Nor test site. The events are very sparsely recorded with only several stations located within 1600 km. We have utilized a regional distance seismic waveform method fitting long-period, complete, three-component waveforms jointly with first-motion observations from regional stations and teleseismic arrays for the moment tensor (MT). The combination of long-period waveforms and first-motion observations provides unique discrimination of these sparsely recorded events. We demonstrate through a series of Jackknife tests of station geometry, and sensitivity analyses that the source-type of the explosions is well constrained. One event, a 1996 Lop Nor shaft explosion displaces large Love waves and reversed Rayleigh waves at one station indicative of a large F-factor due to tectonic release. We show that in this case the combination of long-period waveforms and P-wave first motions discriminate the event source-type. We further demonstrate the sensitivity of network sensitivity solutions to models of tectonic release and tensile damage over a range of F-factor and K-factor, and investigate the free-surface vanishing traction effect in MT estimation. The results demonstrate a robust source type discrimination capability using seismic moment tensor inversion of long- period, regional distance, complete waveforms under sparse recording conditions. We also present an application of a continuous scanning method to small events recorded locally., The original document contains color images.