Ramsey-type theorems

TL;DR: In this paper, the authors present a survey of the general set systems for geometric discrepancy bounds in a more general setting, with a focus on the L 1-discrepancy for halfplanes.

TL;DR: If a graph has bounded clique number and sufficiently large chromatic number, what can the authors say about its induced subgraphs?

TL;DR: This work answers the question in the affirmative that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least and establishes several Ramsey type results for tournaments.

TL;DR: In this paper, the interval number of a graph G is defined as the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i(G).

TL;DR: It is proved that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℽ there is one set within which all the distances are realised.

TL;DR: It is proved that g(n) < n^@a for some @a < 2.

TL;DR: A probability theorem, due to Lovasz, is used to derive lower bounds for various Ramsey functions and a short proof of the known result R(3, t) ⩾ ct 2 ( ln t) 2 is given.