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I'm currently trying to replicate a Carhart four-factor model on the European stock market for a project, but I'm unsure how the factor portfolios are formed (replicating it by using the original approach by Fama and French), more specifically the breakpoints used.

In their 1993 (Common risk factors in the returns on stocks and bonds) paper they write:

We also break NYSE, Amex, and NASDAQ stocks into three book-to-market equity groups based on breakpoints for the bottom 30% (Low) middle 40% (Medium and top 30% (High) for the ranked values of BE/ME for NYSE stocks.

Which I understand as using splitting the stocks into three groups with the same amount of stocks in top and bottom portfolio, but they also write later on:

We use NYSE breakpoints for ME and BE/MEe to allocate [stocks] to five size quintiles... because we use NYSE breakpoints to form the the 25 size-BE/ME portfolios, the portfolios in the smallest size quintile have the most stocks.. Together the five portfolios in the largest ME quintile average about 74% of total value.

Which suggests the portfolio should include the stocks with 30% of the market cap in one portfolio (small amount of stocks) and another with bottom 30% (very large amount of stocks).

Can anyone clarify on the breakpoints being used?

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The Fama/French (1993) paper is based on the widely used CRSP database, maintained by the University of Chicago's Booth School of Business. It provides data for NYSE-, AMEX-, and NASDAQ-listed securities from December 31, 1925 through the present.

The CRSP database

However, in the beginning only NYSE stocks are included in the database. In July 1962, 834 AMEX stocks are added to the sample. The vast majority of NASDAQ stocks are added in December 1972. The addition of AMEX stocks raised the CPI-adjusted value of all stocks in the CRSP database from 2.15 trillion USD to 2.45 trillion USD. Therefore, these stocks tend to have very small market capitalization. Similarly, NASDAQ stocks entering in Dec. 1972 counted for only 12.6% of the total CPI-adjusted market capitalization of the entire CRSP database stocks.

Calculating Breakpoints

The breakpoints in Fama/French (1993) are calculated using only NYSE-stocks. Then, all stocks are sorted into portfolios based on these breakpoints. As the AMEX-, and NASDAQ-stocks are smaller in terms of market capitalization, the smallest portfolio contains the most stocks.

Why did Fama/French (1993) only consider NYSE-stocks for calculating breakpoints?

If the breakpoints were based on considering all stocks in the CRSP sample, the result would be that the breakpoints effectively serve to separate the NYSE stocks from AMEX- and NASDAQ-stocks. Though their approach guarantees, that an equal amount of NYSE stocks is provided in each portfolio. As a result, they avoid the bias of separating their portfolios nearly perfectly on the stock exchange listing. Regardless of which set of stocks is used to form the breakpoints, a large portion of AMEX and NASDAQ stocks end up in portfolios comprised of low market capitalization. That is what they wrote in their Fama/French (1992) paper on p.430:

If we used stocks from all three exchanges to determine the ME breakpoints, most portfolios would include only small stocks after 1973, when NASDAQ stocks are added to the sample.

References

Bali/Engle/Murray (2016), Empirical Asset Pricing: The cross-section of stock returns, 1. ed.

Fama/French (1992), The Cross-Section of stock returns, The Journal of Finance 47(2)

Fama/French (1993), Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33(1)

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