Tommy Walker Mackay | Searcharxiv

arXiv2026

Sharp bounds for covering with large cliques and independent sets

Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k, k) \geq 4k-O(k^\frac{2}{3})}$, and conjectured that $n(k,k)=4k-4$. We prove this conjecture, and also establish the optimal lower bound in the more general case where $k_1$ and $k_2$ are arbitrary. We further consider the generalisation of the problem to $r$-edge-coloured complete graphs in which every vertex is contained in a size-$k$ monochromatic clique of each colour, and obtain upper and lower bounds on the size of such graphs.

math.CO↗

arXiv2026

A Bayesian Gamma-power-mixture survival regression model: predicting the recurrence of prostate cancer post-prostatectomy

In a dataset of 423 patients who had had radical prostatectomy for localised prostate cancer we estimated the apparent Shannon information (ASI) about time to biochemical recurrence in various subsets of the available pre-op variables using a Bayesian Gamma-power-mixture survival regression model. In all the subsets examined the ASI was positive with posterior probability greater than 0.975 . Using only age and results of pre-operative blood tests (PSA and biomarkers) we achieved 0.232 (0.180 to 0.290) nats ASI (0.335 (0.260 to 0.419) bits) (posterior mean and equitailed 95% posterior confidence intervals). This is more than double the mean posterior ASI previously achieved on the same dataset by a subset of the current authors using a log-skew-Student-mixture model, and is greater than that previous value with posterior probability greater than 0.99 . Additionally using pre- or post-operative Gleason grades, operative findings, clinical stage, and presence or absence of extraprostatic extension or seminal vesicle invasion did not increase the ASI extracted. However removing the blood-based biomarkers and replacing them with either pre-operative Gleason grades or findings available from MRI scanning greatly reduced the available ASI to respectively 0.077 (0.038 to 0.120) and 0.088 (0.045 to 0.132) nats (both less than the values using blood-based biomarkers with posterior probability greater than 0.995). A greedy approach to selection of the best biomarkers gave TGFbeta1, VCAM1, IL6sR, and uPA in descending order of importance from those examined.

stat.AP↗