Resolving a problem raised by Norin in 2020, we show that for each k ? N $k \in \mathbb {N}$ , the minimal f ( ) $f(k) with property every graph G $G$ chromatic number at least + 1 $f(k)+1$ contains subgraph H $H$ both connectivity and $k$ satisfies ? 7 \leqslant 7k$ . This result is best-possible up to multiplicative constants, sharpens earlier results of Alon–Kleitman–Thomassen–Saks–Seymour f...

We study a min-max relation conjectured by Saks and West: For any two posets P and Q the size of a maximum semiantichain and the size of a minimum unichain covering in the product P ×Q are equal. For positive we state conditions on P and Q that imply the min-max relation. However, we also have an example showing that in general the min-max relation is false. This disproves the Saks-West conject...

The k-majority game is played with n numbered balls, each coloured with one of two colours. It is given that there are at least k balls of the majority colour, where k is a fixed integer greater than n/2. On each turn the player selects two balls to compare, and it is revealed whether they are of the same colour; the player’s aim is to determine a ball of the majority colour. It has been correc...

We study a min-max relation conjectured by Saks and West: For any two posets P and Q the size of a maximum semiantichain and the size of a minimum unichain covering in the product P ×Q are equal. For positive we state conditions on P and Q that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several...

Quantitative research has dominated applied litigation research, but it seems to lack the flexibility needed to link pretrial research to ongoing courtroom events. Participant observation is a methodology which seems more suitable for studying the dynamic environment of a trial. A 6-day civil trial is used to evaluate participant observation reports against pretrial survey analysis and trial si...

The quantum adversary method is a versatile method for proving lower bounds on quantum algorithms. It yields tight bounds for many computational problems, is robust in having many equivalent formulations, and has natural connections to classical lower bounds. A further nice property of the adversary method is that it behaves very well with respect to composition of functions. We generalize the ...

We prove new results on evasiveness of monotone graph properties by extending the techniques of Kahn, Saks, and Sturtevant [Combinatorica, 4 (1984), pp. 297–306]. For the property of containing a subgraph isomorphic to a fixed graph, and a fairly large class of related n-vertex graph properties, we show evasiveness for an arithmetic progression of values of n. This implies a 1 2 n2 −O(n) lower ...