Hardin and Taylor proved that any function on the reals—even a nowhere continuous one—can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman, who provided upper and lower frontiers (in the subgroup lattice of $\mathrm{Homeo}^+(\mathbb {R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder’s Theorem (that every Archimedean group is abelian).

Theories of early cooperation in human society often draw from a small sample of ethnographic studies of surviving populations of hunter–gatherers, most of which are now sedentary. Borneo hunter–gatherers (Punan, Penan) have seldom figured in comparative research because of a decades-old controversy about whether they are the descendants of farmers who adopted a hunting and gathering way of life. In 2018 we began an ethnographic study of a group of still-nomadic hunter–gatherers who call themselves Punan Batu (Cave Punan). Our genetic analysis clearly indicates that they are very unlikely to be the descendants of neighbouring agriculturalists. They also preserve a song language that is unrelated to other languages of Borneo. Dispersed travelling groups of Punan Batu with fluid membership use message sticks to stay in contact, co-operate and share resources as they journey between rock shelters and forest camps. Message sticks were once widespread among nomadic Punan in Borneo, but have largely disappeared in sedentary Punan villages. Thus the small community of Punan Batu offers a rare glimpse of a hunting and gathering way of life that was once widespread in the forests of Borneo, where prosocial behaviour extended beyond the face-to-face community, facilitating successful collective adaptation to the diverse resources of Borneo's forests.

We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino–Usuba relating stationary reflection to a version of Strong Chang’s Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.

We prove a variation of Easton’s lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2ω ≤ ω2 is consistent with the existence of an ω1-distributive nowhere c.c.c. forcing poset of size ω1. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2(H(θ)), for sufficiently large cardinals θ, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss [13].

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:

1. (1) If ${\cal I}$ is a normal ideal on $\omega _2 $ which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal;

2. (2) For any $n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal ${\cal I}$ on $\omega _n $ which satisfies projective antichain catching, yet ${\cal I}$ is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

Biological reference points (BRPs) in fisheries policy are typically sensitive to stock assessment model assumptions, thus increasing uncertainty in harvest decision-making and potentially blocking adoption of precautionary harvest policies. A collaborative management strategy evaluation approach and closed-loop simulation modelling was used to evaluate expected fishery economic and conservation performance of the sablefish (Anoplopoma fimbria) fishery in British Columbia (Canada), in the presence of uncertainty about BRPs. Comparison of models derived using two precautionary harvest control rules, which each complied with biological conservation objectives and short-term economic objectives given by industry, suggested that both rules were likely to avert biomass decline below limit BRPs, even when stock biomass and production were persistently overestimated by stock assessment models. The slightly less conservative, industry-preferred harvest control rule also avoided short-term economic losses of c. CAN$ 2.7–10 million annually, or 10–50% of current landed value. Distinguishing between the role of BRPs in setting fishery conservation objectives and operational control points that define harvest control rules improved the flexibility of the sablefish management system, and has led to adoption of precautionary management procedures.

We obtain lower bounds for the consistency strength of fully nonregular ultrafilters on ω2.

I felt compelled to write this brief essay after glancing through “The Teacher” section of the recent issues of PS. There are several articles exhorting young educators, soon-to-be educators, or those considering pursuing a career in academia to think about what it means to get in front of a classroom and impart that information which we believe will make a world of difference to our students. These eminent scholars with years of cumulative experience offer sage advice, relevant anecdotes, and a rosy perspective on how exploring alternative avenues of information dissemination will make the classroom experience more positive and fruitful for instructor and student alike.