## View abstract

### Session I.3 - Graph Theory and Combinatorics

Monday, June 12, 14:00 ~ 15:00

## Exponential improvement on diagonal ramsey numbers

### Marcelo Campos

#### University of Oxford, England   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakffb90a2945e8cb3995a2e38b2a01dbec').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyffb90a2945e8cb3995a2e38b2a01dbec = 'm&#97;rc&#101;l&#111;c&#97;mp&#111;s2806' + '&#64;'; addyffb90a2945e8cb3995a2e38b2a01dbec = addyffb90a2945e8cb3995a2e38b2a01dbec + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textffb90a2945e8cb3995a2e38b2a01dbec = 'm&#97;rc&#101;l&#111;c&#97;mp&#111;s2806' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakffb90a2945e8cb3995a2e38b2a01dbec').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyffb90a2945e8cb3995a2e38b2a01dbec + '\'>'+addy_textffb90a2945e8cb3995a2e38b2a01dbec+'<\/a>';

The Ramsey Number $R(k)$ is the minimum $n$ such that every red/blue coloring of the edges of $K_n$ contains a monochromatic $K_k$. In this talk I will discuss a recent work where we show that $R(k)\leq (4-c)^k,$ for some $c \gt 0$. This is the first exponential improvement on the Erd\H{o}s and Szekeres upper bound proved in 1935.

Joint work with Simon Griffiths (PUC - Rio, Brazil), Robert Morris (IMPA, Brazil) and Julian Sahasrabudhe (University of Cambridge, England).