Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

Outcomes of autoconjunctival grafting for primary pterygia when performed by consultant compared with trainee ophthalmologists

Purpose: To define rates of recurrence and surgical complications of primary pterygia excision with autoconjunctival grafting when the surgery is performed by consultant ophthalmologists compared with trainee ophthalmologists. Methods: A total of 174 patients with primary pterygia treated by excision and autoconjunctival grafting were included for analysis. Patients were divided into two groups according to whether their surgery was performed by a consultant ophthalmologist (group A) or a trainee ophthalmologist (group B). Data were collected with respect to demographics, surgical complications and recurrence. Recurrence rates were analysed utilizing Fisher's exact test. Additionally, Kaplan-Meier survival curves for interval censored data were constructed. Surgical complications were analysed utilizing Fisher's exact test. Results: The recurrence rate in group A was 6.8% and in group B was 19.4%. This difference was statistically significant (P = 0.05). The rate of surgical complication occurring in group A was 6.6% and in group B was 23.3% and this was also statistically significant (P = 0.005). No relationship was found between either patient age or the size of pterygium and recurrence or complications in either group A or group B. Conclusion: Autoconjunctival grafting is regarded as the gold standard for preventing pterygium recurrence following excision. This study suggests that the experience of the surgeon can influence success rates and complications. There is a significant learning curve indicating the need to supervise trainee surgeons. © 2006 The Authors; Journal compilatio