Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

Given a family of graphs F , we consider the F -saturation game. In this game, two players alternate adding edges to an initially empty graph on n vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in F . The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let satg (F ; n) denote the number of edges that are in the final graph when both players play optimally.The {C 3 }-saturation game was the first saturation game to be considered, but as of now the order of magnitude of satg({C 3 }, n) remains unknown. We consider a variation of this game. Let C 2k+1 := {C 3 , C 5 , . . . , C 2k+1 }. We prove that satg (C 2k+1 ; n) ≥ ( 1 4 − ǫ k )n 2 + o(n 2 ) for all k ≥ 2 and that satg (C 2k+1 ; n) ≤ ( 1 4 − ǫ ′ k )n 2 + o(n 2 ) for all k ≥ 4, with ǫ k < 1 4 and ǫ ′ k > 0 constants tending to 0 as k → ∞. In addition to this we prove satg({C 2k+1 }; n) ≤ 4 27 n 2 + o(n 2 ) for all k ≥ 2, and satg (C∞ \ C 3 ; n) ≤ 2n − 2, where C∞ denotes the set of all odd cycles.

Let $H$ and $F$ be hypergraphs. We say $H$ {\em contains $F$ as a trace} if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ contains a subhypergraph isomorphic to $F$. In this paper we give an upper bound on the number of edges in a $3$-uniform hypergraph that does not contain $K_{2,t}$ as a trace when $t$ is large. In particular, we show that $$\lim_{t\to \infty}\lim_{n\to \infty} \frac{\mathrm{ex}(n, \mathrm{Tr}_3(K_{2,t}))}{t^{3/2}n^{3/2}} = \frac{1}{6}.$$ Moreover, we show $\frac{1}{2} n^{3/2} + o(n^{3/2}) \leqslant \mathrm{ex}(n, \mathrm{Tr}_3(C_4)) \leqslant \frac{5}{6} n^{3/2} + o(n^{3/2})$.

In simple card games, cards are dealt one at a time and the player guesses each card sequentially. We study problems where feedback (e.g. correct/incorrect) is given after each guess. For decks with repeated values (as in blackjack where suits do not matter) the optimal strategy differs from the "greedy strategy" (of guessing a most likely card each round). Further, both optimal and greedy strategies are far too complicated for real time use by human players. Our main results show that simple heuristics perform close to optimal.

A permutation π is ballot if, for all k, the word π 1 · · · π k has at least as many ascents as it has descents. Let b(n) denote the number of ballot permutations of order n, and let p(n) denote the number of permutations which have odd order in the symmetric group Sn. Callan conjectured that b(n) = p(n) for all n, which was proved by Bernardi, Duplantier, and Nadeau.We propose a refinement of Callan's original conjecture. Let b(n, d) denote the number of ballot permutations with d descents. Let p(n, d) denote the number of odd order permutations with M (π) = d, where M (π) is a certain statistic related to the cyclic descents of π. We conjecture that b(n, d) = p(n, d) for all n and d. We prove this stronger conjecture for the cases d = 1, 2, 3, and d = ⌊(n − 1)/2⌋, and in each of these cases we establish formulas for b(n, d) involving Eulerian numbers and Eulerian-Catalan numbers.