(If someone has a version in higher resolution, or pictures of the conference, please contact John Steel, or myself.)

A very incomplete key, possibly with mistakes:

First row: ?, Diego Rojas-Rebolledo, ?, Leo Harrington, Ernest Schimmerling.

Second: Peter Koekpe; Alexandra, Hugh, and Christine Woodin; Xianghui Shi, me, John Clemens.

Third: John Steel, Alessandro Andretta, Tony Martin.

Fourth: Stevo Todorcevic, Paul Corazza, Philip Welch, Ilijah Farah, Qi Feng, ?, Martin Zeman, Robert Solovay, Richard Laver, Erik Closson (?), James Cummings.

Fifth: ?, Itay Neeman, Thomas Jech, Greg Hjorth, Joan Moschovakis (?), Yiannis Moschovakis, Matthew Foreman (?), Ted Slaman, Jindra Zapletal, Joan Bagaria.

Sixth: Benedikt Löwe, ?, Jean Larson, Bill Mitchell, ?, Carlos di Prisco, ?, Mike Oliver, ?, Lorenz Halbeisen, Derrick Duboise, Peter Koellner.

Seventh, etc: Herb Enderton, ?, ?, Joel Hamkins, Alain Louveau, Slawomir Solecki, ?, Mack Stanley, ?, ?, Tomek Bartoszynski, Paul Larson, Lisa Marks, Richard Ketchersid. ?

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[…] this picture a few days ago, when looking at the old photos from the Martin Conference. I posted here the group picture from that conference. John Steel should be posting the other pictures soon (well, […]

The argument you are looking for is given in Kanamori's book, see Theorem 28.15. For the more nuanced version of the lemma, see section 7D in Moschovakis's descriptive set theory book (particularly 7.D.5-8), or section 3.1 in the Koellner-Woodin chapter of the Handbook.

This problem is very much open. Cheng Yong calls Harrington's $\star$ the assumption that there is a real $x$ such that all $x$-admissible ordinals are $L$-cardinals. From the work of Yong we know that Second- and even Third-order arithmetic do not suffice to prove that Harrington's $\star$ implies the existence of $0^\sharp$. Whether this was poss […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.) If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, t […]

This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR5,151d. (Anyway, it is probably easier to read a more modern presentation, […]

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A proof can be found in Kanamori's book The higher infinite. The result follows from work of Solovay on the theory of uniform indiscernibles and work of Martin on projective scales. Look at sections 14, 28, and 30 of the book. A different approach (using the infinite partition properties of $\omega_1$ and $\omega_2$, themselves due to Solovay and Martin […]

Forcing with sufficiently homogeneous forcing that adds reals is enough to obtain the negation of $(*)$. The point is that if a formula $\phi$ defines a parameter-free well-ordering of $\mathbb R$, then for any ordinal $\alpha$, the statement "$x$ is the $\alpha$-th real in the well-ordering defined by $\phi$" uniquely characterizes $x$ in terms of […]

Assuming that $\gamma$ is finite, the argument is fairly simple: Suppose $\beta\to(\alpha)^\gamma_\delta$, and fix a bijection $f$ between $|\beta|$ and $\beta$. Consider a coloring $c:[|\beta|]^\gamma\to\delta$. Using $f$ , this gives us a coloring $c':[\beta]^\gamma\to\delta$ (here we used that $\gamma$ is finite). We want to argue that there is a $c$ […]

I’ll try to post a key over the next few days. [

Edit:Added, though terribly incomplete.][…] this picture a few days ago, when looking at the old photos from the Martin Conference. I posted here the group picture from that conference. John Steel should be posting the other pictures soon (well, […]