ABSTRACT
The set-coloring Ramsey number Rr,s(k) is defined to be the minimum n such that if each edge of the complete graph Kn is assigned a set of s colors from {1, ,r}, then one of the colors contains a monochromatic clique of size k. The case s=1 is the usual r-color Ramsey number, and the case s=r-1 was studied by Erdos, Hajnal and Rado in 1965, and by Erdos and Szemerédi in 1972. The first significant results for general s were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that Rr,s(k)=2Θ(kr) if s/r is bounded away from 0 and 1. In the range s=r-o(r), however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) coloring, and use it to determine Rr,s(k) up to polylogarithmic factors in the exponent for essentially all r, s, and k.
ABSTRACT
PURPOSE: To compare global health-related quality of life (HRQoL) and overall survival (OS) in patients with head and neck cancer treated with intensity modulated radiation therapy (IMRT), conformal radiation therapy (3DCRT) or conventional radiation therapy (2DRT). METHODS AND MATERIALS: In this real-world, multi-institutional and prospective study, HRQoL outcomes were assessed using the European Organisation for Research and Treatment of Cancer Quality-of-life Questionnaire Core 30 (EORTC QLQ-C30) and European Organisation for Research and Treatment of Cancer Quality-of-life Questionnaire Head and Neck 43 (H&N43) questionnaires. Item response theory was used to generate a global HRQoL score, based on the 71 questions from both forms. The effect of treatment modality on HRQoL was studied using multivariate regression analyses. Survival was estimated using the Kaplan-Meyer method, and groups were compared by the log-rank test. RESULTS: Five hundred and seventy patients from 13 institutions were included. Median follow-up was 12.2 months. Concerning the radiation technique, 29.5% of the patients were treated with 2DRT, 43.7% received 3DCRT, and 26.8% were treated with IMRT. A higher proportion of patients receiving 2DRT had a treatment interruption of more than 5 days (69% vs 50.2% for 3DCRT and 42.5% for IMRT). IMRT had a statistically significant positive effect on HRQoL compared with 3DCRT (ß= 2.627, standard error = 0.804, P = .001) and 2DRT had a statistically significant negative effect compared with 3DCRT (ß= -5.075, standard error = 0.926, P < .001). Patients receiving 2DRT presented a worse OS (P = .01). There were no differences in OS when IMRT was compared with 3DCRT. CONCLUSIONS: IMRT provided better HRQoL than 3DCRT, which provided better HRQoL than 2DRT. Patients receiving 2DRT presented a worse OS, which might be related to more frequent treatment interruptions.
