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Abstract A$$(p,q)$$-colouring of a graph$$G$$is an edge-colouring of$$G$$which assigns at least$$q$$colours to each$$p$$-clique. The problem of determining the minimum number of colours,$$f(n,p,q)$$, needed to give a$$(p,q)$$-colouring of the complete graph$$K_n$$is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers$$r_k(p)$$. The best-known general upper bound on$$f(n,p,q)$$was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where$$p=q$$have been obtained only for$$p\in \{4,5\}$$, each of which was proved by giving a deterministic construction which combined a$$(p,p-1)$$-colouring using few colours with an algebraic colouring. In this paper, we provide a framework for proving new upper bounds on$$f(n,p,p)$$in the style of these earlier constructions. We characterize all colourings of$$p$$-cliques with$$p-1$$colours which can appear in our modified version of the$$(p,p-1)$$-colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying$$(p,p)$$-colourings, which would otherwise make this problem intractable for large values of$$p$$. In addition, we generalize our algebraic colouring from the$$p=5$$setting and use this to give improved upper bounds on$$f(n,6,6)$$and$$f(n,8,8)$$.
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This content will become publicly available on April 2, 2026
Numerical solution of a PDE arising from prediction with expert advice
Abstract This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by$$n$$experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($$n\leq 10$$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.
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- Award ID(s):
- 1944925
- PAR ID:
- 10608972
- Publisher / Repository:
- Cambridge Press
- Date Published:
- Journal Name:
- European Journal of Applied Mathematics
- ISSN:
- 0956-7925
- Page Range / eLocation ID:
- 1 to 27
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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